text:
# Counting cancer cells with computer vision for time-lapse microscopy

Some people characterize TheEGG as a computer science blog. And although (theoretical) computer science almost always informs my thought, I feel like it has been a while since I have directly dealt with the programming aspects of computer science here. Today, I want to remedy that. In the process, I will share some Python code and discuss some new empirical data collected by Jeff Peacock and Andriy Marusyk.[1]

Together with David Basanta and Jacob Scott, the five of us are looking at the in vitro dynamics of resistance to Alectinib in non-small cell lung cancer. Alectinib is a new ALK-inhibitor developed by the Chugai Pharmaceutical Co. that was approved for clinical use in Japan in 2014, and in the USA at the end of 2015. Currently, it is intended for tough lung cancer cases that have failed to respond to crizotinib. Although we are primarily interested in how alectinib resistance develops and unfolds, we realize the importance of the tumour’s microenvironment, so one of our first goals — and the focus here — is to see how the Alectinib sensitive cancer cells interact with healthy fibroblasts. Since I’ve been wanting to learn basic computer vision skills and refresh my long lapsed Python knowledge, I decided to hack together some cell counting algorithms to analyze our microscopy data.[2]

In this post, I want to discuss some of our preliminary work although due to length constraints there won’t be any results of interest to clinical oncologist in this entry. Instead, I will introduce automated microscopy to computer science readers, so that they know another domain where their programming skills can come in useful; and discuss some basic computer vision so that non-computational biologists know how (some of) their cell counters (might) work on the inside. Thus, the post will be methods heavy and part tutorial, part background, with a tiny sprinkle of experimental images.[3] I am also eager for some feedback and tips from readers that are more familiar than I am with these methods. So, dear reader, leave your insights in the comments.

### Automated microscopy

A few years ago, I wrote about droids taking academic jobs from a typical theorist’s focus on automating literature review and writing. However, the real automation breakthroughs are on the experimental side. There used to be a time when people would have to look through microscopes by eye, count cells manually, and sketch what they see. I think I even did that at some point in school. Now, Jeff can prepare a 96-well plate and put it in a machine that will take precise photos of the same exact positions in each well, every 2 hours.[4] Six days later, I can ssh into the same machine and get about 51 thousand images — 40 GB of TIFs — to analyze.

However, even under hundredfold magnification, cells are hard to see and tell apart. So a number of physical and biological tricks are used before the images are even recorded. The first of these is phase-contrast. In my grade-school experience with microscopes, I had a light source under the sample, it would shine through it, with some of the light being absorbed (or otherwise obstructed) by the cell, and thus creating darker patches for me to identify as life. This is called bright-field microscopy, and only takes advantage of the amplitude of the light — that which makes things brighter or darker to our eyes and cameras. But light carries a second channel: phase. Phase is shifted by passing through the different media of the cell membrane and other components, with the amount of shift depending on thickness and refractive index of the object. This information is then captured by the digital camera, combined with the standard bright field information, and converted into an optical thickness for each pixel. The result is a phase-contrast image that is much clearer than standard bright-field and that allows us to see details that would normally only be visible through staining of dead cells.

But even if you can see the cells, it is still often impossible to tell them apart because different cell types can be morphologically identical. To overcome this, a biological engineering trick can be combined with fluorescence imaging. We can take our separate cell lines — in our case: parental cancer cells, resistant cancer cells, and fibroblasts — and add an extra piece of DNA to code for a fluorescing (but otherwise non-functional) protein. Then, during image capture, we shine a light of the excitatory wavelength that is absorbed by the protein and then re-emitted at a longer wavelength which is recorded (after the source wavelength is physically filtered out) by the digital camera. Thus, instead of looking at obstructions (and phase-shift) of an external light source, we end up using the protein itself as the light source and recording where that light is brightest. By giving different cell lines different proteins that are excited by and emit different wavelengths, we can then tell those cells apart by the wavelength of their fluorescence.

The practical result is that for each snapshot of part of a well,[5] we end up with three photos: one phase-contrast, and one color channel for each of the two fluorescent proteins. The programming problem becomes transforming these 3 photos into a measurement of size for the populations of cancer cells and fibroblasts.

An example of the 3 channels that are recorded from microscopy. From left to right: phase contrast, GFP, mCherry. All data is intensity, but I used the green and red colour space for the 2nd and 3rd image to make them easier to distinguish at a glance.

In our experimental system, we created two variants of each of our cancer and fibroblast cell lines by inserting the genes coding for GFP (absorption 395 nm, emission 509 nm) into one and mCherry (587/610 nm) into the other. These genes are inserted throughout the genome, and so we expect them to be expressed and thus to fluoresce whenever proteins are being transcribed. Further, our fluoroproteins are not confined to any specific part of the cell and distribute themselves haphazardly throughout the cell. This allows the fluorescent area to serve as a good unit of size for our populations.

Conveniently — and not at all by coincidence — fluorescent area is much easier to measure than the number of cells. Mostly, this is due to the tendency of the cancer cells to clump, making segmentation difficult, and the lanky and faint fibroblasts making it hard to find their boundaries.[6] Of course, this means that as we eventually transform this into replicator dynamics, we will have be mindful of our units being fluorescent area and not cell count, but that choice of units makes no difference to the mathematics.

### Basic image processing

“If you want to make an apple pie from scratch,” said Carl Sagan in Cosmos (1980), “you must first invent the universe.” This is equally true for programming. If I want to build my analysis code ‘from scratch’ then I can go arbitrarily far back to progressively more basic building blocks. At some point, if I don’t want to ‘invent the universe’, I have to pick a set of packages to start from. I decided to start from the Python equivalent of store-bought crust and canned apple pie filling: the OpenCV package (and numpy, of course).[7]

import numpy as np
import cv2

OpenCV makes file operations with images extremely simple:

def AreaCount(col_let, row_num, fot_num, dirName = '', ret_img = True):
head = dirName + col_let + str(row_num) + '-' + str(fot_num)

if ret_img:

In this case, the outputs of the cv2.imread functions are simply two-dimensional arrays of intensities. Although the microscope itself creates different kinds of TIFs for the phase-contrast and fluorescent photos, so the output arrays require a conversion into a common 8-bit format:[8]

#switch to 8bit with proper normalizing
if np.amax(rdImage) > 255:
rdImage8 = cv2.convertScaleAbs(rdImage, alpha = (255.0/np.amax(rdImage)))
else:
rdImage8 = cv2.convertScaleAbs(rdImage)

if np.amax(gnImage) > 255:
gnImage8 = cv2.convertScaleAbs(gnImage, alpha = (255.0/np.amax(gnImage)))
else:
gnImage8 = cv2.convertScaleAbs(gnImage)

To go beyond file operations, thought, I need a brief detour into the real apples of OpenCV.

Straight out of the box, OpenCV gives me tools like histogram equalization for enhancing image contrast. Histogram equalization for contrast works on the premise that what makes an image high contrast is a maximization of information conveyed by the intensity of each pixel. In other words, we want to stretch out the intensity histogram into the highest-entropy version: a uniform distribution. This is a relatively straightforward process, just take the cumulative distribution for the pixels, and then create a function such that a pixel of value $k \in {0, U}$ with $F(k) = \text{Pr}[x \leq k]$ maps to a new value $k' = U F(k)$, that way in the new image $\text{Pr}[x' \leq k'] \approx \frac{k'}{U}$. Not a hard function to implement for yourself, but with OpenCV it is just one line: img_hicon = cv2.equalizeHist(img).

An example of histogram equalization that makes obvious the colour gradient present in the mCherry channel. On the right of each image is the histogram for their intensity distribution, with the raw histogram in red and the cumulative distribution as a black line. For the equalized image, I am using a different color map with blue corresponding to low intensity and red to high intensity. In the bottom histogram, I am also including bigger bins to make it clearer how the distribution is moved towards a uniform one.

What you might notice right away is the colour gradient radiating outwards. This is not an issue in the phase-contrast image, but the fluorescence images have a glow around their outside, like reverse vignetting,[9] that is not apparent to the eye until you turn up the contrast (or try to threshold).

With this problem and tool in mind, let me walk through the simple function FluorescentAreaMark that sits at the base of my current image processing. To get rid of the above glow I use a variant of the method we used to notice it: contrast limited adaptive histogram equalization.

def FluorescentAreaMark(img):
#1: contrast limited adaptive histogram equalization to get rid of glow
clahe = cv2.createCLAHE(clipLimit=50.0, tileGridSize=(20,20))

clahe_img = clahe.apply(img)

mCherry channel after processing with adaptive histogram equalization. On the left is the image with blue corresponding to low intensity and red to high intensity, and on the right is the histogram of intensities with the cumulative distribution in black.

The algorithm works by dividing the image into 400 tiles (20 tiles by 20 tiles), and normalizing the contrast in each one. Each pixel is then given a value by bilinear interpolation between the 4 tiles nearest it. This gives a sharp distinction of the cells from their background, and we can then detect them with a simple threshold, followed by some noise canceling[10]:

#2: threshold and clean out salt-pepper noise
ret, thresh_img = cv2.threshold(clahe_img,127,255, cv2.THRESH_BINARY)

kernel = np.ones((3,3),np.uint8)

clean_img = cv2.morphologyEx(thresh_img, cv2.MORPH_OPEN, kernel, iterations = 2)

return clean_img

The threshold in line 11 simply converts the 8bit image to a binary one, by setting all pixels with intensity lower than 127 in the original image to 0 in the final image and those with intensity of 127 and higher to 1. The noise filtering step in line 15 is morphological opening — a fancy name but a straightforward algorithm. Its two basic components are erosion and dilation, both require a kernel that defines a local neighbourhood. In our case the kernel in line 13 is the one-distance Moore neighbourhood. Erosion is an AND operation on the binary mask: for a focal pixel, if every pixel in its local neighbourhood is 1 in the original image than that pixel is a 1 in the final image, else it is zero. Dilation is an OR operation: if at least one pixel in the local neighbourhood of the focal pixel is 1 in the original image then the focal pixel is a 1 in the final image. Morphological opening cycles erosion followed by dilation a number of times given by iterations — two in our case. Thus, lonely pixels that are most likely due to noise are eliminated, but the overall area of the non-noise pixels is not decreased.

The returned clean_img on line 17 is then a binary two dimensional array with a ‘1’ for each fluorescent pixel and ‘0’ otherwise. If we want the total fluorescent area then we just sum this array, outside a small buffer area around the edges which tends to contain more mistakes in correcting the edge glow. If desired, we can also combine the raw phase-contrasts images with the two processed fluorescent areas for a final visualization of the distribution of fluorescence throughout the spatially structured population:

rdFA = FluorescentAreaMark(rdImage8)
gnFA = FluorescentAreaMark(gnImage8)

ign_buf = 30 #how big of an edge do we ignore?
rd_area = (rdFA[ign_buf:-ign_buf,ign_buf:-ign_buf] > 0)
gn_area = (gnFA[ign_buf:-ign_buf,ign_buf:-ign_buf] > 0)

#create image to save
img_out = []
if ret_img:
bW = 0.85

img_out = cv2.merge((

return [np.sum(rd_area),np.sum(gn_area)], img_out

Distribution of fluorescent area. Non-small cell lung cancer cells are marked with mCherry in red, and non-cancerous fibroblasts are marked GFP in green. Colours in this image are a binary mask after the processing from this post.

In the near future, I plan to use this sort of spatial distribution information to learn about operationalizing the local environment. For the next post, however, I will just look at the overall population dynamics estimated by this method and what we can learn from them. Although the basics of computer vision that I hacked together in this post are sufficient for our purposes with this project, it does violate my ambitions as a computer scientist. If there is time and interest then I will also write some future posts to refine this code to minimize categorization errors and introduce some proper cell segmentation.

### Notes

1. I was also inspired by Jan Poleszczuk’s Compute Cancer blog to start posting actual code snippets to allow for easier collaboration and engagement with programmers interested in biology, or biologists looking to start programming. I recommend checking out his blog for many code snippets and tricks across various languages. For example, if you want to oppose my choice of Python in this post then check out his speed comparison of Java, C++, MATLAB, Julia or Python for cellular automatons post for ammunition. Also, let me know what you think of interspersing code within this post. If there is a lot of interest in this format then I will try to include it in more future posts.
2. I am, of course, reinventing the wheel. A lot of great packages and programs already exist for processing and analyzing microscopy data, and we even have some in-house code here at the IMO. However, I wanted to go through the motions of this as a learning exercise. This is to make sure that I understood exactly what the algorithm is doing, and to make sure that it is a good fit for the particulars of our experimental set up.
3. Having several target audiences means that the first section on microscopy is probably too basic for biologists, and the second section on computer vision is probably too basic for programmers. So feel free to jump around. As with many of my posts, this is also meant as a tutorial for future Artem, for when I return to this experiment and code after a few months and have no idea what it does, why it exists, and how to get it running. I will try to confine the more pedantic parts of this documentation to footnotes.
4. In fact, as automation continues its perpetual march forward, even the preparation step is being eliminated. Labs like the Foundry at Imperial College London are automating the whole bench biology pipeline. I wonder if this will soon allow us to program in vitro experiments from basic components and then run them in remote automated labs, much like how we currently send in silico simulations to clusters. My optimistic hope is that the communication with machines forced by automation will also encourage more and more experimental biology graduate students to learn more and more computer science. Maybe that exposure to CS will allow for a flourishing of algorithmic biology. My cynical side, however, wonders at the disconnect between programming — a skill that everyone, from any field, should pick up — and the more niche aspects of algorithmic thinking or theoretical computer science.
5. At our magnification in this post, the field of view for a single photo is about 1/5th of the diameter of the well. The microscope takes photos of three different positions — the same positions at every 2-hour time step — in each well.
6. These limitations of not segmenting single cells can be overcome with more work. I will have to return to them in future blogposts if I want to look at single-cell tracking and lineage tracing.
7. Python has packages for almost anything, and usually in many versions, so there are plenty of alternatives to OpenCV. I chose it due to its great documentation, and from chatting with Chandler Gatenbee — the IMO expert on computer vision — who also builds his code with this package. Chandler’s warning was that the hardest part of OpenCV is getting it installed.

I use the Anaconda distribution of Python 3, and it is pretty easy to get it with the built-in package manager from binstar: conda install -c https://conda.binstar.org/menpo opencv. It is also a good idea to have a separate environment for opencv to work in, to avoid breaking dependencies. This can be done with conda create -n opencv numpy scipy scikit-learn matplotlib python=3 and then accessed later from command line (before running jyputer notebook, for example) with source activate opencv. For a slightly more detailed tutorial, see this.

The code I discuss in this post is available on my GitHub, and the numbered lines of code correspond to the numbers in the first commit FluorescentArea.py. The images through out this post were generated from the jyputer notebook in which I hacked the code together and paint. It is straightforward to generate these images with OpenCV, but if you want the details then ask me in the comments and I will post some more code snippets.

8. The function cv2.converScaleAbs tends to clip-down the intensities, setting every pixel above 255 to 255, so if there are intensities above 255 in the 16-bit images, it is better to first renormalize (as is done in lines 30,35 by setting alpha to something other than 1) to make the brightest pixel 255. Note that this renormalization means that I cannot make physical sense of the raw intensity numbers and have to rely on contrast. I could use a consistent renormalization to allow using raw intensity numbers, but since the rest of the processing focuses on contrast, there is no reason to worry about it.
9. We are not sure what physical process causes this glow. The apparent circular symmetry makes me think it has something to do with the lens. Andriy suspects it has more to do with partial fluorescence from the well walls, and to remedy this, we are using a different type of 96 well plate in follow up experiments.

From the Department of Computer Science at Oxford University and Department of Translational Hematology & Oncology Research at Cleveland Clinic, I marvel at the world through algorithmic lenses. My mind is drawn to evolutionary dynamics, theoretical computer science, mathematical oncology, computational learning theory, and philosophy of science. Previously I was at the Department of Integrated Mathematical Oncology at Moffitt Cancer Center, and the School of Computer Science and Department of Psychology at McGill University. In a past life, I worried about quantum queries at the Institute for Quantum Computing and Department of Combinatorics & Optimization at University of Waterloo and as a visitor to the Centre for Quantum Technologies at National University of Singapore. Meander with me on Google+ and Twitter.

### 12 Responses to Counting cancer cells with computer vision for time-lapse microscopy

1. Abel Molina says:

An interesting alternative way to get rid of the glow might be to compute a baseline for each square of the 20×20 grid, and then subtract from each pixel. This is motivated by thinking about the corner case where there are no target cells in one of the squares of the 20 x 20 grid – with the current method there might be false positives then (though maybe this is very very very very unlikely, of course : ) )

• Thanks for the comment.

I didn’t really talk about the ‘CL’ part of CLAHE, but I think part of it’s goal is to stop this from happening. If the transformation is too drastic. If any bin(s) in the histogram is very very high, as would be for the empty space case, it distributes the excess above the clipping limit across all the bins, and thus makes the resultant map displace those points less. Also, since all of this is run through thresholding after, even if that was missing, it would probably turn into salt-pepper style noise, due to the random variance in the background, which I think try to clean up with morphological open.

But I think you are totally on the right track, though, with just engaging with histograms for each square of the grid directly and with simpler transformations. I think the way to do your baseline approach would be to compute the median of the image before processing and then adjust the pixels in each square of the grid so that the medians match. As long as the squares remain relatively large with respect to the size of single-cells, this shouldn’t cause too much of a problem.

The above would be less destructive that complete histogram equalization. But HE kind of does this already (since it moves all the medians to be 127), so this would just be HE without the stretching.

Focusing on the median would be especially good for the phase-contrast images — which I didn’t discuss much in this post. There even cell presence will not interfere with the algorithm because the cells tend to make exponential decays in the histogram around the median (about same levels of phase up and down shifting usually happens, compared to the plate — I guess this means that optical thickness should be measured as deviation from the median).

• And of course, Python with all its import antigravity makes this super easy to do:

med = np.median(img)

It also saves us having to do bilinear interpolation, but maybe that will make it slow. I am not sure how efficient medianBlur is for big images, compared to doing interpolation.

• This interesting, but would it not be easier to subtract an empty well from the image? Are there any disadvantages to this as you see it (besides asking the biologist to leave one of the wells empty, which is already done oftentimes on the edges)?

2. For footnote 1: I definitely like the code snippets, especially for posts like these.

3. I was also wondering why 8 bit is necessary, just for computational speed or do some of the methods get confused with different values?

• A number of OpenCV methods are built only for 8-bit. So yes, this is for efficiency since re-coding those methods from scratch would be a waste of time and slower in Python than the C++ implementation of OpenCV. I also think that in most cases 16-bit is overkill, it gives us a false sense of sensitivity not actually present in the apparatus. I can believe our microscope can relatively reliably tell apart 256 intensities… but 65,000+ seems like too precise for our sort of experiments.

As to your other question on subtracting an empty well. I have tried this, it is one of the reasons that we occasionally filmed the empty side-wells that Andriy likes to leave. However, from my experience there does not seem to be enough consistency in the background noise between wells to make this worthwhile. Some of it might depend on the geometry of the lens (my initial hope) but a lot also depends on scatter from other wells and random effects like condensation on the well’s glass barrier. The CLAHE seems to do a better job of eliminating these effects, but of course, it is important to tune for each experimental set up.

If you’ve been playing around more with these sort of things and want to contribute a guest post, I think people might find it interesting. I know that I would like to learn more.


text:
• anonymous
Which of the following would complete the truth table shown? T, T, T, F T, F, T, T T, T, F, F
Mathematics

Looking for something else?

Not the answer you are looking for? Search for more explanations.

format: plain


text:
2 deleted 197 characters in body

As Ryan Budney points out in a comment, there's not much you can say without specifying the slice disk $D$. If you have an explicit presentation of $D$ as a ribbon disk, then you can easily turn this into an explicit handle diagram for $D^4 \setminus D$. If you cut the ribbon self-intersections of the ribbon disk, the result is an unlink in $S^3$, and this unlink gives the 1-handles of the diagram (in Kirby notation, dotted circles). For each ribbon, add a 0-framed 2-handle which links each of the adjacent dotted circles.

As far as I know, every knot that is known to be (smoothly) slice is also known to be a ribbon knot (i.e. there are no candidate counterexamples to the "slice = ribbon" conjecture for knots).

If you have a slice disk $D$ which is not a ribbon disk, then the associated Morse function on $D$ will have local maxima, and each such maximum will correspond to a 3-handle in a generalization of the above recipe for a handle diagram.

1

As Ryan Budney points out in a comment, there's not much you can say without specifying the slice disk $D$. If you have an explicit presentation of $D$ as a ribbon disk, then you can easily turn this into an explicit handle diagram for $D^4 \setminus D$. If you cut the ribbon self-intersections of the ribbon disk, the result is an unlink in $S^3$, and this unlink gives the 1-handles of the diagram (in Kirby notation, dotted circles). For each ribbon, add a 0-framed 2-handle which links each of the adjacent dotted circles.

As far as I know, every knot that is known to be (smoothly) slice is also known to be a ribbon knot (i.e. there are no candidate counterexamples to the "slice = ribbon" conjecture for knots).

If you have a slice disk $D$ which is not a ribbon disk, then the associated Morse function on $D$ will have local maxima, and each such maximum will correspond to a 3-handle in a generalization of the above recipe for a handle diagram.


text:
# Abstracts

## Objective:

to analyze the effectiveness of a support intervention on the burden and stress of family caregivers and on the stroke survivors’ independence level, compared to the Control Group.

## Method:

a quasi-experimental study conducted with 37 participants (Intervention Group, n=20; and Control Group, n=17). The intervention lasted 8 months. The outcomes of the caregivers (burden and stress) and of the survivors (independence level) were measured by the Zarit, Perceived Stress and Katz scales, at the following moments: pre-intervention, the fourth month of the intervention and post-intervention. The differences of these outcomes between groups and intra-group and the effect size were calculated using the Mann-Whitney and Friedman tests (Bonferroni adjustment by Wilcoxon) and the Kendall’s W coefficient.

## Results:

the Intervention Group reduced burden (p=0.039) and stress (p=0.009), mainly, after 8 months of intervention, which was not observed in the Control Group. The independence level did not change between the groups or moments (p>0.05). The intervention presented moderate effect size (p=0.45 and p=0.54).

## Conclusion:

the intervention was effective to reduce the burden and stress of family caregivers, but did not alter the stroke survivors’ independence level, when compared to the Control Group.

Descriptors:
Evidence-Based Nursing; Caregivers; Family; Stroke; Program; Social Support

## Objetivo:

analisar a efetividade de uma intervenção de apoio na sobrecarga e no estresse de cuidadores familiares e no grau de independência dos sobreviventes de Acidente Vascular Cerebral, em comparação ao Grupo Controle.

## Método:

estudo quase-experimental realizado com 37 participantes (Grupo Intervenção n=20 e Grupo Controle n=17). A intervenção durou 8 meses. Os desfechos dos cuidadores (sobrecarga e estresse) e dos sobreviventes (grau de independência) foram mensurados pelas escalas de Zarit, Estresse Percebido e Katz, nos momentos: pré-intervenção, quarto mês da intervenção e pós-intervenção. As diferenças desses desfechos intergrupos e intragrupo e o tamanho do efeito foram calculados pelos testes de Mann-Whitney e de Friedman (ajuste de Bonferroni pelo Wilcoxon) e o coeficiente de Kendall W.

o Grupo Intervenção reduziu a sobrecarga (p=0,039) e o estresse (p=0,009), principalmente, após 8 meses de intervenção, o que não foi observado no Grupo Controle. O grau de independência não se alterou entre os grupos ou momentos (p>0,05). A intervenção apresentou tamanho de efeito moderado (p=0,45 e p=0,54).

## Conclusão:

A intervenção foi efetiva para reduzir a sobrecarga e o estresse dos cuidadores familiares, mas não alterou o grau de independência dos sobreviventes de Acidente Vascular Cerebral, quando comparada ao Grupo Controle.

Descritores:
Enfermagem Baseada em Evidências; Cuidadores; Família; Acidente Vascular Cerebral; Programa; Apoio Social

## Objetivo:

analizar la efectividad de una intervención de apoyo sobre la sobrecarga y el estrés de los cuidadores familiares y sobre el grado de independencia de los sobrevivientes de accidente cerebrovascular, en comparación con el Grupo de Control.

## Método:

estudio cuasi experimental realizado con 37 participantes (Grupo de Intervención n = 20 y Grupo de Control n = 17). La intervención duró 8 meses. Los resultados de los cuidadores (sobrecarga y estrés) y sobrevivientes (grado de independencia) se midieron mediante las escalas de Zarit, Estrés Percibido y Katz, en los momentos: preintervención, cuarto mes de intervención y posintervención. Las diferencias entre estos resultados intergrupales e intragrupales y el tamaño del efecto se calcularon mediante las pruebas de Mann-Whitney y Friedman (ajuste de Bonferroni por Wilcoxon) y el coeficiente de Kendall W.

el Grupo de Intervención redujo la sobrecarga (p = 0,039) y el estrés (p = 0,009), principalmente después de ocho meses de intervención, lo que no se observó en el Grupo de Control. El grado de independencia no varió entre grupos o momentos (p>0,05). La intervención tuvo un tamaño de efecto moderado (p=0,45 y p=0,54).

## Conclusión:

la intervención fue eficaz para reducir la sobrecarga y el estrés de los cuidadores familiares, pero no alteró el grado de independencia de los sobrevivientes de ACV, en comparación con el Grupo de Control.

Descriptores:
Enfermería Basada en la Evidencia; Cuidadores; Familia; Accidente Cerebrovascular; Programa; Apoyo Social

# Introduction

Stroke is one of the main causes of chronic cognitive and functional impairments in adults and aged individuals worldwide. The rehabilitation of stroke survivors is generally long; the family, or a single family caregiver, assumes home care, with the execution of complex tasks and without due preparation, which results in physical, economic, emotional and psychosocial burden(11 Cheng HY, Chair SY, Chau JPC. Effectiveness of a strength-oriented psychoeducation on caregiving competence, problem-solving abilities, psychosocial outcomes and physical health among family caregiver of stroke survivors: A randomised controlled trial. Int J Nurs Stud Adv. 2018;87:84-93. doi: http://doi.org/10.1016/j.ijnurstu.2018.07.005
http://doi.org/10.1016/j.ijnurstu.2018.0...
).

In the absence of a source of social support, especially health services and health, this burden can increase and jeopardize both the health of the caregiver and the quality of care provided to the patient. This can result in the emergence or worsening of stress in the caregiver and in the difficulty recovering the autonomy of the stroke survivors to carry out the everyday basic activities(22 Caro CC, Costa JD, Cruz DMC. Burden and Quality of Life of Family Caregivers of Stroke Patients. Occup Ther Health Care. 2018;32(2):154-71. doi: http://doi.org/10.1080/07380577.2018.1449046
http://doi.org/10.1080/07380577.2018.144...
).

These negative consequences for the caregiverstroke survivor dyad show the need for reducing such burden. Thus, the literature recommends enabling to these caregivers multicomponent interventions that emphasize care for their own health, while providing continuous training for the care of the stroke survivor(33 Bakas T, McCarthy M, Miller T. An Update on the State of the Evidence for Stroke Family Caregiver and Dyad Interventions. Stroke. 2017;48:e122-e25. doi: http://doi.org/10.1161/STROKEAHA.117.016052
http://doi.org/10.1161/STROKEAHA.117.016...
).

Nurses can be at the forefront among the health professionals that are part of formal social support(44 Day CB, Bierhals CCBK, Mocellin D, Predebon M, Santos NO, Pizzol FLFD, et al. Nursing Home Care Intervention Post Stroke (SHARE) 1 year effect on the burden of family caregivers for older adults in Brazil: A randomized controlled trial. Health Soc Care Community. 2020;00:1-10. doi: https://doi.org/10.1111/hsc.13068
https://doi.org/10.1111/hsc.13068...
), establishing partnerships with other members of the multidisciplinary team in the implementation of strategies or multicomponent interventions to provide support, education and advice to the caregivers, either individually or collectively.

These interventions must be developed considering the caregivers’ needs(55 Silva JK, Anjos KF, Santos VC, Boery RNSO, Santa Rosa DO, Boery EN. Interventions for stroke survivor caregivers: a systematic review. Rev Panam Salud Publica. 2018;42:e114. doi: https://doi.org/10.26633/RPSP.2018.114
https://doi.org/10.26633/RPSP.2018.114...
) and the recovery phase of the stroke survivors, which is divided into acute phase (occurring between the transition period from hospital discharge to the first six months in the community) and the chronic phase (starting from six months after returning to the community)(33 Bakas T, McCarthy M, Miller T. An Update on the State of the Evidence for Stroke Family Caregiver and Dyad Interventions. Stroke. 2017;48:e122-e25. doi: http://doi.org/10.1161/STROKEAHA.117.016052
http://doi.org/10.1161/STROKEAHA.117.016...
).

Although the caregiver’s burden remains throughout the care period(66 Pesantes MA, Brandt LR, Ipince A, Miranda JJ, DiesCanseco F. An exploration into caring for a stroke-survivor in Lima, Peru: Emotional impact, stress factors, coping mechanisms and unmet needs of informal caregivers. eNeurologicalSci. 2017;6:33-50. doi: http://doi.org/10.1016/j.ensci.2016.11.004/
http://doi.org/10.1016/j.ensci.2016.11.0...
), there are more publications of intervention studies in the acute phase, which reveals a need for more interventionist research studies in the chronic phase(33 Bakas T, McCarthy M, Miller T. An Update on the State of the Evidence for Stroke Family Caregiver and Dyad Interventions. Stroke. 2017;48:e122-e25. doi: http://doi.org/10.1161/STROKEAHA.117.016052
http://doi.org/10.1161/STROKEAHA.117.016...
), since the caregivers need constant guidelines on elementary and complex care, knowing that problems can emerge during the care period.

Most of the research studies on support interventions after stroke, focusing on the caregivers, come from highincome countries. In Brazil, the interventions to reduce the burden on the family caregiver family caregivers of survivors of this disease based on scientific evidence are scarce(55 Silva JK, Anjos KF, Santos VC, Boery RNSO, Santa Rosa DO, Boery EN. Interventions for stroke survivor caregivers: a systematic review. Rev Panam Salud Publica. 2018;42:e114. doi: https://doi.org/10.26633/RPSP.2018.114
https://doi.org/10.26633/RPSP.2018.114...
), especially in the chronic phase of stroke. In this sense, interventions with low financial cost and easy development can be viable in health care, especially in the public sector, so as to supplement routine care or as an alternative to it, usually provided by the public and private health services.

As the burden generally has relation with the caregiver’s stress and with the stroke survivor’s independence level(77 Kobayasi DY, Rodrigues PR, Fhon SJ, Silva LM, Souza AC, Chayamiti CE. Overload, social support network and emotional stress on the caregiver of elderly. Av Enferm. 2019;37(2):140-8. doi: https://doi.org/10.15446/av.enferm.v37n2.73044
https://doi.org/10.15446/av.enferm.v37n2...
-88 Santana MS, Oliveira DL, Santos MM, Rangel RL, Chaves RN, Reis LA. Overload of the family caregivers of the elderly with functional dependence. Rev Kairós, 2018:21(1):337-53. doi: https://doi.org/10.23925/2176-901X.2018v 21i1p 337-353
https://doi.org/10.23925/2176-901X.2018v...
), the following hypothesis will be tested: There will be differences in the burden and stress of family caregivers and in the independence level of stroke survivors in the Intervention Group, when compared to the Control Group. Therefore, the objective of the study was to analyze the effectiveness of a support intervention on the burden and stress of family caregivers and on the stroke survivors’ independence level compared to the Control Group.

# Method

## Study design

A prospective and quasi-experimental study, based on the Transparent Reporting of Evaluations with Nonrandomized Designs (TREND) statement, which represents a supplement to the Consolidated Standards of Reporting Trials (CONSORT) statement(99 Des Jarlais DC, Lyles C, Crepaz N, the TREND Group. Improving the reporting quality of nonrandomized evaluations of behavioral and public health interventions: The TREND statement. Am J Public Health. 2004;94:361-6. doi: http://doi.org/10.2105/ajph.94.3.361
http://doi.org/10.2105/ajph.94.3.361...
).

Randomization is a process that distributes a “random” number of participants in different groups (experimental and control) in order to preserve the characteristics that similar between them. For a study to be classified as randomized, it is necessary that all the randomization stages (allocation sequence generation, concealment and implementation mechanism) are rigorously performed(1010 Moher D, Hopewell S, Schulz K, Montori V, Gøtzsche PC, Devereaux PJ, et al. ConSoRT 2010 explanation and elaboration: updated guidelines for reporting parallel group randomised trials. BMJ. 2010;340:c869. doi: http://doi.org/10.1136/bmj.c869
http://doi.org/10.1136/bmj.c869...
).

Thus, this study was not classified as randomized because allocation sequence generation occurred by means of the only alternative and acceptable method called minimization, which, although not eliminating bias in all the known and unknown factors, allows for the balance between the Intervention and Control groups, by selecting the participants’ factors(1010 Moher D, Hopewell S, Schulz K, Montori V, Gøtzsche PC, Devereaux PJ, et al. ConSoRT 2010 explanation and elaboration: updated guidelines for reporting parallel group randomised trials. BMJ. 2010;340:c869. doi: http://doi.org/10.1136/bmj.c869
http://doi.org/10.1136/bmj.c869...
). The choice of this method is recommended for smaller groups and was necessary due to the lethality of stroke, which could further reduce the sample.

The study was approved by the Research Ethics Committee of the State University of Southwest Bahia under CAAE 71341017.5.0000.0055, therefore observing Brazilian Resolution number 466 of 2012 and the Declaration of Helsinki. All the participants authorized the research by signing the Free and Informed Consent Form in two copies, one being filed by one of the researchers (first author) and the other remaining in possession of the caregivers.

## Participants and sample size

The sample was for convenience and the study took place in the city of Guanambi, Bahia (BA), Brazil. The family caregivers were recruited between September 2017 and March 2018, in their homes. Therefore, it was necessary to identify the stroke survivors in the chronic phase of the disease that had an attendance register from 2014, either in the regional hospital unit (n=152) or in the 17 Basic Health Units (BHUs) (n=69) managed by the Unified Health system (Sistema Único de Saúde, SUS).

The eligibility criteria for the caregivers were as follows: identifying themselves as the primary family caregivers of a stroke survivor who was in the chronic phase and presented care-related dependence in one or more functions (assessed by the Katz scale), time working as a caregiver between 6 and 50 months, being 18 years old or more, not receiving any financial remuneration for the care provided, living in urban areas and living in the same household as the stroke survivor or not.

## Intervention

The intervention consisted in supporting the family caregivers. Therefore, it was divided into two parts (individual and group) and lasted 8 months, combining the terms “acquisition of skills” and “education”.

Before starting the intervention, the lead researcher (first author) conducted a previous study that investigated the needs of all the family caregivers, regardless of their future allocations in the Intervention Group (IG) or Control Group (CG). The results showed that these caregivers needed education in health about stroke and about daily care to the survivors, having free time and getting assistance for their physical and mental health(1111 Silva JK, Angos KF, Alves JP, Rosa DOD, Boery RNSO. Needs for Family Caregivers of Cerebrovascular Accident survivors. Invest Educ Enferm. 2020;38(3):e06. doi: https://doi.org/10.17533/udea.iee.v38n3e06
https://doi.org/10.17533/udea.iee.v38n3e...
). Knowing these needs supported the organization of the group and individual components of the intervention.

For preparing the group component, during the planning of the intervention, health professionals were selected using the following inclusion criteria: minimum experience of 1 year in caregiver and stroke survivor assistance and in higher education teaching in the areas of Medicine, Nursing, Psychology, Physiotherapy, Physical Education or Nutrition. To ensure adherence of these professionals to the study protocol, the lead researcher maintained regular contact according to the time availability of these employees, with a mean of three individual meetings (before starting the intervention) to elaborate the content of the themes and their respective informative booklets.

The execution of the group component of the support intervention took place in the auditorium of a public university for 8 months (one monthly session during two hours), with the participation of the previously selected health professionals and of the IG caregivers. The 8 sessions took place in the form of thematic conversation circles and were guided by Paulo Freire’s theoretical and methodological framework on the pedagogy of autonomy, adapted to the study context(1212 Freire, P. Pedagogia da autonomia: saberes necessários à prática educativa. São Paulo: Editora Paz e Terra; 1996.).

An action plan was developed for all the sessions containing information relating to their respective stages: problematization (description of the problem to be discussed), topic, date, time, locus, objectives, methodological strategies (presentation of the proposal, video, dynamics, group discussion, etc.), food for the participants, human and material resources, evaluation of each session and references used for each theme. Each session was recorded with a video camera by a research assistant and the following themes were covered: 1) the importance of self-care; 2) general guidelines on stroke (concept, types, risk factors and referral in the health care network); 3) negative and positive aspects of care; 4) care with nutrition in the dyad; 5) stress coping mechanisms in the caregiver; 6) positioning of the stroke survivor and caregiver and ergonomic posture of the caregiver; 7) deterioration in the health of the dyad and death of the person being cared for and 8) body care and body hygiene after stroke. Sessions one, three, five and seven started with a 30-minute relaxation exercise led by a Physical Education professional and sequentially mediated by a Psychologist, while the other sessions were conducted, respectively, by graduates in Medicine, Nutrition, Physiotherapy and Nursing. An informative booklet was handed out at the end of each session with the content of the theme addressed.

In the individual component, Physiotherapy and Psychology consultations were conducted by health professionals working in the health laboratory of a University Center and who did not participate in the group component. A total of 10 Physiotherapy sessions were availed, as well as unlimited Psychological care sessions until the end of the intervention, although the frequency of this service was related to the individual health needs of each caregiver.

The family caregivers allocated in the CG and in the IG were instructed to maintain their routine care in the public and/or private health units and, therefore, should not interrupt the usual care link with these services available to them, after accepting to participate in the study. The Control Group did not receive care from any health professional who participated in the support intervention.

To maintain adherence to the study, all the caregivers underwent routine laboratory tests offered by the SUS before the intervention, namely: blood tests [blood count, total cholesterol and lipoproteins, triglycerides, glycaemia, glutamic oxaloacetic transaminase (GOT), glutamic pyruvic transaminase (GPT), gamma glutamyl transferase (Gamma GT), hormone thyroidstimulating (HTS), thyroxine free flowing blood (T4 free), urea, creatinine and uric acid]; urinalysis (Type I urinalysis for dosage of the Sediment Abnormal Elements) and stool parasitologic test. Subsequently, the lead researcher delivered the results to the caregivers, offered verbal guidelines based on the SUS protocols and referred them to the BHUs for follow-up. Transportation was paid for the IG caregivers who lived far from the locus of the group sessions and a telephone contact was made during the previous week and another the day before the intervention to confirm the date, time and place of the session.

## Instruments and outcomes

A form containing variables about the caregiver and the stroke survivor was developed for this study, based on previous surveys(11 Cheng HY, Chair SY, Chau JPC. Effectiveness of a strength-oriented psychoeducation on caregiving competence, problem-solving abilities, psychosocial outcomes and physical health among family caregiver of stroke survivors: A randomised controlled trial. Int J Nurs Stud Adv. 2018;87:84-93. doi: http://doi.org/10.1016/j.ijnurstu.2018.07.005
http://doi.org/10.1016/j.ijnurstu.2018.0...
-22 Caro CC, Costa JD, Cruz DMC. Burden and Quality of Life of Family Caregivers of Stroke Patients. Occup Ther Health Care. 2018;32(2):154-71. doi: http://doi.org/10.1080/07380577.2018.1449046
http://doi.org/10.1080/07380577.2018.144...
,44 Day CB, Bierhals CCBK, Mocellin D, Predebon M, Santos NO, Pizzol FLFD, et al. Nursing Home Care Intervention Post Stroke (SHARE) 1 year effect on the burden of family caregivers for older adults in Brazil: A randomized controlled trial. Health Soc Care Community. 2020;00:1-10. doi: https://doi.org/10.1111/hsc.13068
https://doi.org/10.1111/hsc.13068...
,77 Kobayasi DY, Rodrigues PR, Fhon SJ, Silva LM, Souza AC, Chayamiti CE. Overload, social support network and emotional stress on the caregiver of elderly. Av Enferm. 2019;37(2):140-8. doi: https://doi.org/10.15446/av.enferm.v37n2.73044
https://doi.org/10.15446/av.enferm.v37n2...
-88 Santana MS, Oliveira DL, Santos MM, Rangel RL, Chaves RN, Reis LA. Overload of the family caregivers of the elderly with functional dependence. Rev Kairós, 2018:21(1):337-53. doi: https://doi.org/10.23925/2176-901X.2018v 21i1p 337-353
https://doi.org/10.23925/2176-901X.2018v...
). The primary outcome was the caregiver’s burden in relation to the care provided to the stroke survivor. To assess it, the Zarit Burden Interview scale was used, consisting in 22 items that assess health condition, psychological and financial situation, interpersonal relationships and social and personal life. The total score of the scale varied from 0 to 88, where the higher the score, the greater the burden. Although it was developed for caregivers of older adults and of people with dementia, this scale can be applied to caregivers of people with various mental and physical ailments. In Brazil, its internal consistency and validity were tested in caregivers of older adults with depression and the Cronbach’s alpha value was 0.87(1313 Scazufca M. Brazilian version of the Burden Interview scale for the assessment of burden of care in carers of people with mental illnesses. Braz J Psychiatry. 2002;24(1):12-7. doi: http://dx.doi.org/10.1590/S1516-44462002000100006
http://dx.doi.org/10.1590/S1516-44462002...
).

The secondary outcomes included the family caregivers’ stress and the stroke survivor’s independence level. The perceived stress scale was proposed in 1983 and validated for Brazil in the aged population in 2007; it consists of 14 Likert-type questions with answer options ranging from 0 to 4. The scores of the sum of the points obtained in the questions vary from 0 to 56. Higher score indicate more stress in the last 30 days. The reliability of the scale was tested in individuals aged from18 to 70 years old in a population-based study in Brazil and presented a Cronbach’s alpha of 0.775(1414 Faro A. Confirmatory factor analysis of three versions of the Perceived Stress Scale (PSS): a population-based study. Psicol Reflex. Crit. 2015;28(1):21-30. doi: http://dx.doi.org/10.1590/1678-7153.201528103
http://dx.doi.org/10.1590/1678-7153.2015...
).

To identify the stroke survivor’s independence level, the Basic Activities of Daily Living scale was applied to the caregivers. This instrument assesses the functional independence of older adults and of other dependents in six respective functions: feeding, sphincter control, transfer, going to the bathroom, ability to dress and bathe. The score varies from 0 to 6 points, where one point is assigned to each “yes” answer, the individual being considered as follows: independent in all functions (zero score) or dependent in one to six functions. This scale was cross-culturally adapted Brazil in older adults and their caregivers, having its internal consistency tested and approved by Cronbach’s alpha values ranging from 0.80 to 0.92. Its validity was not assessed directly, but the Cronbach’s alpha coefficient levels provided empirical evidence of its validity(1515 Lino VTS, Pereira SRM, Camacho LAB, Ribeiro Filho ST, Buksman S. Cross-cultural adaptation of the Independence in Activities of Daily Living Index (Katz Index). Cad Saúde Pública. 2008;24(1):103-12. doi: https://doi.org/10.1590/S0102-311X2008000100010
https://doi.org/10.1590/S0102-311X200800...
).

All four instruments were applied in the homes of the family caregivers. To verify the suitability of these instruments to the study, a pilot test was performed with 8 caregivers who were not part of the sample (n=37).

## Data collection schedule

Data collection was conducted in-person and for the three outcomes (burden and stress of the caregiver and independence level of the stroke survivor) the application of the instruments took place in three stages: pre-intervention (T0) (between September 2017 and March 2018); during the fourth month of the intervention (T1) and post-intervention (T2) (between December 2018 and February 2019). The lead researcher trained a specialist nurse for collecting these data, with the last application of the scales occurring up to 3 months after the intervention has been completed.

## Allocation and blinding

After collecting data at the pre-intervention moment, a statistician (external to the research) conducted the generation of the participants’ allocation sequence, through the minimization process according to the outcomes, the sociodemographic characteristics of the caregiver and the stroke survivor, seeking homogeneity of these variables in both groups. In this allocation process (minimization), the first individual was randomly allocated and, for each subsequent participant, the treatment allocation which minimized the imbalance of the factors selected from the groups was identified(1010 Moher D, Hopewell S, Schulz K, Montori V, Gøtzsche PC, Devereaux PJ, et al. ConSoRT 2010 explanation and elaboration: updated guidelines for reporting parallel group randomised trials. BMJ. 2010;340:c869. doi: http://doi.org/10.1136/bmj.c869
http://doi.org/10.1136/bmj.c869...
).

Blinding (blinding of the participants’ allocation in the Control and Intervention groups) was possible only for the trained nurse who applied all the instruments and for the statistician who conducted the minimization process and data analysis, since the caregivers and professionals were aware of the intervention.

## Statistical analysis

Medians and interquartile range (IQR), absolute values and percentages were used to respectively describe the continuous and categorical variables related to the caregivers’ characteristics [age, gender, marital status, income (in this study, the minimum wage varied between R$937.00 and R$ 998.00 reais - Brazilian money -, from 2017 to 2019 and the value of the last year was considered, corresponding to the value of the US dollar in 2019, which was R\$ 3.6570), occupation, schooling, kinship, living with the patient, family support, care time and hours, burden and stress] and of the stroke survivor (age, gender and independence level), regardless of their integration into the experimental design. The differences in the median and proportion of these variables between the Intervention and Control groups were analyzed by using the Mann-Whitney test and Fisher’s Chi-square or exact test, respectively.

To assess the differences of the outcome variables analyzed (burden and stress of the caregiver; independence level of the stroke survivor) between the groups (i.e., Control and Intervention) at the same time, the Mann-Whitney test was used. For the intragroup analysis at the different moment (T0, T1 and T2), the Friedman tests were used (with Bonferroni adjustment by Wilcoxon). Effect size calculation was performed for the comparisons between groups (i.e., comparisons of the differences between T0, T1 and T2), Kendall’s W coefficient being adopted as the effect size indicator. Kendall’s W coefficient assumes that values equal to or near to 0 indicate a negligible effect size and that values close or equal to 1 indicate large effect(1616 Tomczak M, Tomczak EWA. The need to report effect size estimates revised. An overview of some recommended measures of effect size. Trends Sport Sci. [Internet]. 2014 [cited 2020 Oct 15];21(1):19-25. Available from: https://www.researchgate.net/publication/303919832_The_need_to_report_effect_size_estimates_revisited_ An_overview_of_some_recommended_measures_of_effect_size
https://www.researchgate.net/publication...
). The Statistical Package for Social Sciences (SPSS) software, version 21.0, was used for all the statistical analyses, considering a significance level of 5%.

# Results

The results are presented according to the TREND statement. Of the 44 participants eligible for the study, 37 consented to participate, being assigned to the Intervention (n=20) and Control (n=17) groups.

Table 1
Descriptive variables of the caregivers (n=37), stroke survivors (n=37) and the respective primary and secondary outcomes, with equivalence between the Intervention (n=20) and Control (n=17) groups. Guanambi, BA, Brazil, 2017-2019
Figure 1
Flowchart of the study participants. Guanambi, BA, Brazil, 2017-2019

The baseline descriptive variables of the caregivers, stroke survivors and primary (caregiver burden) and secondary (caregiver’s stress and Stroke survivor’s independence level) outcomes in the Intervention and Control groups were tested for homogeneity, both groups being statistically similar. Most of the caregivers in the study were women, married, with low financial income, and unemployed. Of these, more than half lived in the same house that the stroke survivors, were their children or spouses and provided care several hours of the day, regardless of the stroke survivor’s gender. Despite the homogeneity of the samples, the IG presents a higher median age, schooling level, family support and stroke survivors with a greater degree of functional impairment, as shown in Table 1.

Table 2
Comparison between the scores of the primary and secondary outcomes in the Intervention (n=16) and Control (n=16) groups at the pre-intervention, during the intervention and post-intervention moments. Guanambi, BA, Brazil, 2017-2019

Regarding the family caregivers’ adherence to the protocol of this study, it is noteworthy that all the patients underwent the laboratory tests. For the IG, the number of sessions in which the caregivers participated varied from 4 (n=3) to 6 or more (n=13) and, of these, only 7 people accepted transportation payment for the group component of the intervention. Of the 10 caregivers who required psychological care, 6 did not start the sessions because of the difficulty of scheduling by telephone contacts, one reported that the group service was sufficient to improve emotional support and three caregivers attended only 1-4 sessions, as they mentioned that the group meetings offered better psychological well-being. Of the 12 caregivers assessed for the physiotherapy sessions, 8 attended between 6 and 10 sessions and the absences were justified by: increased blood pressure levels (n=1) and lack of family support on the scheduled dates and times (n=3) (data not recorded in table).

The median burden, stress and independence level were not significantly different (p>0.05) in the Control and Intervention groups at any of the three moments. On the other hand, the moments assessed differed significantly for two caregiver variables studied in the IG. For the caregiver burden variable, a reduction in the median between the moments was evidenced, ranging from 33.50 at T0 to 25.00 at T2. For the caregiver stress variable, there was also a reduction in the median between the moments, mainly ranging from 29.00 at T0 to 15.50 at T2. On the contrary, the stroke survivor’s independence level did not present any difference between the moments, maintaining a constant median of 4.00. The results also point to a moderate effect size on the outcome variables of the caregiver, which shows the effectiveness of the support intervention over the proposed time, since the Intervention Group presented a difference over time, which was not the case in the Control Group (Table 2).

# Discussion

Considering the mean effect size found in this study, the results indicate that a support intervention, centered on the caregivers, can reduce the burden and stress from care, but still cannot improve the independence of stroke survivors in the chronic phase over time. The outcome variables analyzed are multifactorial and, therefore, if other interventions were added to this, they may result in an even greater positive effect for the participants.

Most of the other multicomponent interventions that provided social support to family caregivers occurred during the acute phase of stroke, as this period is marked by the apex of burden and stress from care. For these aforementioned studies, the main modality used was (psycho)educational, through the individual instrumentalization to the caregiver, about care practices and problem solving. However, these studies presented divergent results, with the significant reduction or not of caregiver burden and stress in the Intervention Group(11 Cheng HY, Chair SY, Chau JPC. Effectiveness of a strength-oriented psychoeducation on caregiving competence, problem-solving abilities, psychosocial outcomes and physical health among family caregiver of stroke survivors: A randomised controlled trial. Int J Nurs Stud Adv. 2018;87:84-93. doi: http://doi.org/10.1016/j.ijnurstu.2018.07.005
http://doi.org/10.1016/j.ijnurstu.2018.0...
,1717 Hekmatpou D, Baghban EM, Dehkordi LM. The effect of patient care education on burden of care and the quality of life of caregivers of stroke patients. J Multidiscip Healthc. 2019;12:211-7. doi: http://doi.org/10.2147/JMDH.S196903
http://doi.org/10.2147/JMDH.S196903...
), which suggests the importance of implementing and assessing these interventions over time.

On the other hand, multicomponent interventions conducted in the chronic phase of stroke, with the group modality, related to the educational and skills acquisition measures, showed a significant reduction in the caregivers’ burden and stress, in the only group evaluated. The main effects for the participants were as follows: better support among the caregivers, greater self-confidence and self-care, increased ability to cope with problems and management of negative feelings, incentive to using the community resources and applying knowledge about the care techniques(1818 Christensen ER, Golden SL, Gesell SB. Perceived Benefits of Peer Support Groups for Stroke Survivors and Caregivers in Rural North Carolina. N C Med J. 2019;80(3), 143-8. doi: https://doi.org/10.18043/ncm.80.3.143
https://doi.org/10.18043/ncm.80.3.143...
-1919 Mores G, Whiteman RM, Ploeg J, Knobl P, Cahn M, Klaponski L, et al. An Evaluation of the Family Informal Caregiver Stroke Self-Management Program. Can J Neurol Sci. 2018;45(6):660-8. doi: http://doi.org/10.1017/cjn.2018.335
http://doi.org/10.1017/cjn.2018.335...
).

Unlike the aforementioned studies that were developed in the chronic phase of stroke, this support intervention used two groups (Intervention and Control), which reinforces the greater effect of the group strategy in reducing the caregivers’ burden and stress during home care, especially in the psychosocial aspects.

Generally, coexistence in a support group awakens the perception of not being alone and is preferable to individual care. Social support between the group members and the professionals improves the feeling of emotional well-being(2020 Walker MF, Birchall S, Cobley C, Condon L, Fisher R, Fletcher-Smith J, et al. Biopsychosocial intervention for stroke carers (BISC): results of a feasibility randomised controlled trial and nested qualitative interview study. Clin Rehabil. 2020;34(10):1268-81. doi: https://doi.org/10.1177/0269215520937039
https://doi.org/10.1177/0269215520937039...
), mainly in the sharing of successful experiences for problem solving. The participants of this study also made these findings explicit.

Knowing that other people were able to modify their care routine can motivate the caregivers to develop their own autonomy to acquire a new perspective towards the care of the other and self-care, which leads to a release of their distresses and anguishes. As a result, they can present reduced stress and burden.

The extension of the intervention of this study is in line with what the literature recommends, between 5 and 9 sessions(33 Bakas T, McCarthy M, Miller T. An Update on the State of the Evidence for Stroke Family Caregiver and Dyad Interventions. Stroke. 2017;48:e122-e25. doi: http://doi.org/10.1161/STROKEAHA.117.016052
http://doi.org/10.1161/STROKEAHA.117.016...
). In opposition to most of the research studies already published, the caregivers’ individual needs were incorporated. However, focusing this intervention on the caregiver may have little influence on the improvement of the stroke survivors’ results(33 Bakas T, McCarthy M, Miller T. An Update on the State of the Evidence for Stroke Family Caregiver and Dyad Interventions. Stroke. 2017;48:e122-e25. doi: http://doi.org/10.1161/STROKEAHA.117.016052
http://doi.org/10.1161/STROKEAHA.117.016...
), which was also evidenced in this research.

Similarly to this support intervention, another family-centered study, which used a short message service and cell phone calls for 8 weeks, with the objective of improving the basic daily activities of stroke survivors, also failed to increase the participants’ independence in the chronic phase of this disease(2121 Kamwesiga JT, Eriksson GM, Tham K, Fors U, Ndiwalana, A, von Koch L, et al. A feasibility study of a mobile phone supported family-centred ADL intervention, F@ce™, after stroke in Uganda. Global Health. 2018;14:82. doi: https://doi.org/10.1186/s12992-018-0400-7
https://doi.org/10.1186/s12992-018-0400-...
). It is probable that the focus of this support intervention, in responding only the unmet needs of the caregivers, influenced this result. Stroke survivors may also have reached the apex of their motor recovery prior to the study, as it is during the acute phase that the professionals intensify the use of therapies for musculoskeletal strengthening(2222 Guler MA, Erhan B, Yalcinkaya EY. Caregiver burden in stroke inpatients: a randomized study comparing robotassisted gait training and conventional therapy. Acta Neurol Belg. 2020. https://doi.org/10.1007/s13760-020-01465-5
https://doi.org/10.1007/s13760-020-01465...
) and, therefore, after hospital discharge, the survivors can present functional decline. In addition to that, the way in which the family caregivers stimulated the stroke survivors’ mobility and independence at home after receiving guidelines in the group component was not verified in this research. Verifying these care practices can ease understanding of the effect of the intervention on the independence of these survivors for the daily activities.

Most of the recovery of the motor functions occurs until the sixth month post-stroke and, after this period, functional capacity decreases progressively until it stabilizes(2323 Campos LM, Martins BM, Cabral NL, Franco SC, Pontes-Neto OM, Mazin SC, et al. How many patients become functionally dependent after a stroke? A 3-Year population-based study in Joinville, Brazil. PLoS One. 2017;12(1):e0170204. doi: http://doi.org/10.1371/journal.pone.0170204
http://doi.org/10.1371/journal.pone.0170...
). Some factors can interfere with the recovery of these functions, such as the early use of antithrombotic therapy, type of stroke(2323 Campos LM, Martins BM, Cabral NL, Franco SC, Pontes-Neto OM, Mazin SC, et al. How many patients become functionally dependent after a stroke? A 3-Year population-based study in Joinville, Brazil. PLoS One. 2017;12(1):e0170204. doi: http://doi.org/10.1371/journal.pone.0170204
http://doi.org/10.1371/journal.pone.0170...
), its severity at admission and advanced age of the survivors(2424 Rejnö A, Nasic S, Bjälkefur K, Bertholds E, Jood K. Changes in functional outcome over five years after stroke. Brain Behav. 2019;9(6):e01300. doi: http://doi.org/10.1002/brb3.1300
http://doi.org/10.1002/brb3.1300...
), its severity at hospital admission and older age of survivors

The CG participants also maintained the stroke survivors’ independence level and reduced the caregivers’ burden and stress, although the results have not been significant. The performance of laboratory tests and the referral to the health unit may have influenced the CG caregivers in lifestyle change, greater concern for health and search for assistance support, which may have modified the results of this study.

These general data indicate that there is need for a continued intervention to the caregivers and their family members who have survived a stroke, so that there is an improvement in this dyad’s health and well-being over time. Those interventions can be preferably coordinated by nurses in their work spaces, adapting them to the needs of caregivers and patients during the individual home education strategies or in the training of the support groups in the community(2525 Magwood GS, Nichols M, Jenkins C, Logan A, Qanungo S, Zigbuo-Wenzler E, et al. Community-based interventions for stroke provided by nurses and community health workers: A Review of the Literature. J Neurosci Nurs. 2020;52(4):152-9. doi: 10.1097/JNN.000000000000512
https://doi.org/10.1097/JNN.000000000000...
) as light and low financial cost technologies.

Most of the interventions in health developed on the theme of stroke focus on support in the recovery and rehabilitation of survivors of this disease. Few interventions have been directed to the caregiver’s needs(2020 Walker MF, Birchall S, Cobley C, Condon L, Fisher R, Fletcher-Smith J, et al. Biopsychosocial intervention for stroke carers (BISC): results of a feasibility randomised controlled trial and nested qualitative interview study. Clin Rehabil. 2020;34(10):1268-81. doi: https://doi.org/10.1177/0269215520937039
https://doi.org/10.1177/0269215520937039...
), as well as to their effect towards the stroke survivors. In Brazil, this scenario is even more critical, since these interventions in health are still scarce. Therefore, this support intervention that has been developed represents an advance in scientific knowledge and contributes to the clinical practice, as it emerges as an alternative or complementary model to the current methods of daily care in the public or private health services for the reduction of the caregivers’ burden and stress. The greater adherence of the participants in the group component suggests the preference for this approach in relation to the individual component (psychological support). As a care manager and main executor of the health education actions in the various health units, nurses can be in charge of conducting that intervention, mainly in the use of the group component.

In intervention studies, it is not always possible to blind researchers and participants due to the interaction between them, which can generate some bias in the result. However, to reduce this risk, the evaluator and data collector were blinded and a Control Group was used, as recommended by the TREND statement, which represent strengths of this study. In addition to these points, the minimization process was adopted for the participants allocated in the Intervention and Control groups, since it was not possible to perform randomization.

The limitations of the study refer to the sample size and to the non-randomization of the sample, which precludes generalizing the results to other groups and, therefore, the data must be interpreted with caution. Therefore, in future research studies, a larger sample selected by randomization can be used, especially when analyzing sustainability over time. Even so, the findings in this study may help in the design of such research studies.

# Conclusion

The intervention was effective to directly reduce the family caregivers’ burden and stress in the post-stroke chronic phase. However, the repercussion of its effects on stroke survivors was not identified, as they did not present changes in their independence level for the performance of the basic daily activities, when compared to the Control Group.

Such findings reinforce that care to the caregivers is essential to improve their physical and psychological health and should not be limited to the early months of their role. In addition to assessing and incorporating the stroke survivors’ needs, in this intervention, may generate more benefits for these participants.

# Acknowledgments

We thank Professor Rafael Pereira and Icarus José R. Santos for their collaboration in the stage of the statistical analyses.

# References

• 1
Cheng HY, Chair SY, Chau JPC. Effectiveness of a strength-oriented psychoeducation on caregiving competence, problem-solving abilities, psychosocial outcomes and physical health among family caregiver of stroke survivors: A randomised controlled trial. Int J Nurs Stud Adv. 2018;87:84-93. doi: http://doi.org/10.1016/j.ijnurstu.2018.07.005
» http://doi.org/10.1016/j.ijnurstu.2018.07.005
• 2
Caro CC, Costa JD, Cruz DMC. Burden and Quality of Life of Family Caregivers of Stroke Patients. Occup Ther Health Care. 2018;32(2):154-71. doi: http://doi.org/10.1080/07380577.2018.1449046
» http://doi.org/10.1080/07380577.2018.1449046
• 3
Bakas T, McCarthy M, Miller T. An Update on the State of the Evidence for Stroke Family Caregiver and Dyad Interventions. Stroke. 2017;48:e122-e25. doi: http://doi.org/10.1161/STROKEAHA.117.016052
» http://doi.org/10.1161/STROKEAHA.117.016052
• 4
Day CB, Bierhals CCBK, Mocellin D, Predebon M, Santos NO, Pizzol FLFD, et al. Nursing Home Care Intervention Post Stroke (SHARE) 1 year effect on the burden of family caregivers for older adults in Brazil: A randomized controlled trial. Health Soc Care Community. 2020;00:1-10. doi: https://doi.org/10.1111/hsc.13068
» https://doi.org/10.1111/hsc.13068
• 5
Silva JK, Anjos KF, Santos VC, Boery RNSO, Santa Rosa DO, Boery EN. Interventions for stroke survivor caregivers: a systematic review. Rev Panam Salud Publica. 2018;42:e114. doi: https://doi.org/10.26633/RPSP.2018.114
» https://doi.org/10.26633/RPSP.2018.114
• 6
Pesantes MA, Brandt LR, Ipince A, Miranda JJ, DiesCanseco F. An exploration into caring for a stroke-survivor in Lima, Peru: Emotional impact, stress factors, coping mechanisms and unmet needs of informal caregivers. eNeurologicalSci. 2017;6:33-50. doi: http://doi.org/10.1016/j.ensci.2016.11.004/
» http://doi.org/10.1016/j.ensci.2016.11.004/
• 7
Kobayasi DY, Rodrigues PR, Fhon SJ, Silva LM, Souza AC, Chayamiti CE. Overload, social support network and emotional stress on the caregiver of elderly. Av Enferm. 2019;37(2):140-8. doi: https://doi.org/10.15446/av.enferm.v37n2.73044
» https://doi.org/10.15446/av.enferm.v37n2.73044
• 8
Santana MS, Oliveira DL, Santos MM, Rangel RL, Chaves RN, Reis LA. Overload of the family caregivers of the elderly with functional dependence. Rev Kairós, 2018:21(1):337-53. doi: https://doi.org/10.23925/2176-901X.2018v 21i1p 337-353
» https://doi.org/10.23925/2176-901X.2018v21i1p
• 9
Des Jarlais DC, Lyles C, Crepaz N, the TREND Group. Improving the reporting quality of nonrandomized evaluations of behavioral and public health interventions: The TREND statement. Am J Public Health. 2004;94:361-6. doi: http://doi.org/10.2105/ajph.94.3.361
» http://doi.org/10.2105/ajph.94.3.361
• 10
Moher D, Hopewell S, Schulz K, Montori V, Gøtzsche PC, Devereaux PJ, et al. ConSoRT 2010 explanation and elaboration: updated guidelines for reporting parallel group randomised trials. BMJ. 2010;340:c869. doi: http://doi.org/10.1136/bmj.c869
» http://doi.org/10.1136/bmj.c869
• 11
Silva JK, Angos KF, Alves JP, Rosa DOD, Boery RNSO. Needs for Family Caregivers of Cerebrovascular Accident survivors. Invest Educ Enferm. 2020;38(3):e06. doi: https://doi.org/10.17533/udea.iee.v38n3e06
» https://doi.org/10.17533/udea.iee.v38n3e06
• 12
Freire, P. Pedagogia da autonomia: saberes necessários à prática educativa. São Paulo: Editora Paz e Terra; 1996.
• 13
Scazufca M. Brazilian version of the Burden Interview scale for the assessment of burden of care in carers of people with mental illnesses. Braz J Psychiatry. 2002;24(1):12-7. doi: http://dx.doi.org/10.1590/S1516-44462002000100006
» http://dx.doi.org/10.1590/S1516-44462002000100006
• 14
Faro A. Confirmatory factor analysis of three versions of the Perceived Stress Scale (PSS): a population-based study. Psicol Reflex. Crit. 2015;28(1):21-30. doi: http://dx.doi.org/10.1590/1678-7153.201528103
» http://dx.doi.org/10.1590/1678-7153.201528103
• 15
Lino VTS, Pereira SRM, Camacho LAB, Ribeiro Filho ST, Buksman S. Cross-cultural adaptation of the Independence in Activities of Daily Living Index (Katz Index). Cad Saúde Pública. 2008;24(1):103-12. doi: https://doi.org/10.1590/S0102-311X2008000100010
» https://doi.org/10.1590/S0102-311X2008000100010
• 16
Tomczak M, Tomczak EWA. The need to report effect size estimates revised. An overview of some recommended measures of effect size. Trends Sport Sci. [Internet]. 2014 [cited 2020 Oct 15];21(1):19-25. Available from: https://www.researchgate.net/publication/303919832_The_need_to_report_effect_size_estimates_revisited_ An_overview_of_some_recommended_measures_of_effect_size
» https://www.researchgate.net/publication/303919832_The_need_to_report_effect_size_estimates_revisited_An_overview_of_some_recommended_measures_of_effect_size
• 17
Hekmatpou D, Baghban EM, Dehkordi LM. The effect of patient care education on burden of care and the quality of life of caregivers of stroke patients. J Multidiscip Healthc. 2019;12:211-7. doi: http://doi.org/10.2147/JMDH.S196903
» http://doi.org/10.2147/JMDH.S196903
• 18
Christensen ER, Golden SL, Gesell SB. Perceived Benefits of Peer Support Groups for Stroke Survivors and Caregivers in Rural North Carolina. N C Med J. 2019;80(3), 143-8. doi: https://doi.org/10.18043/ncm.80.3.143
» https://doi.org/10.18043/ncm.80.3.143
• 19
Mores G, Whiteman RM, Ploeg J, Knobl P, Cahn M, Klaponski L, et al. An Evaluation of the Family Informal Caregiver Stroke Self-Management Program. Can J Neurol Sci. 2018;45(6):660-8. doi: http://doi.org/10.1017/cjn.2018.335
» http://doi.org/10.1017/cjn.2018.335
• 20
Walker MF, Birchall S, Cobley C, Condon L, Fisher R, Fletcher-Smith J, et al. Biopsychosocial intervention for stroke carers (BISC): results of a feasibility randomised controlled trial and nested qualitative interview study. Clin Rehabil. 2020;34(10):1268-81. doi: https://doi.org/10.1177/0269215520937039
» https://doi.org/10.1177/0269215520937039
• 21
Kamwesiga JT, Eriksson GM, Tham K, Fors U, Ndiwalana, A, von Koch L, et al. A feasibility study of a mobile phone supported family-centred ADL intervention, F@ce™, after stroke in Uganda. Global Health. 2018;14:82. doi: https://doi.org/10.1186/s12992-018-0400-7
» https://doi.org/10.1186/s12992-018-0400-7
• 22
Guler MA, Erhan B, Yalcinkaya EY. Caregiver burden in stroke inpatients: a randomized study comparing robotassisted gait training and conventional therapy. Acta Neurol Belg. 2020. https://doi.org/10.1007/s13760-020-01465-5
» https://doi.org/10.1007/s13760-020-01465-5
• 23
Campos LM, Martins BM, Cabral NL, Franco SC, Pontes-Neto OM, Mazin SC, et al. How many patients become functionally dependent after a stroke? A 3-Year population-based study in Joinville, Brazil. PLoS One. 2017;12(1):e0170204. doi: http://doi.org/10.1371/journal.pone.0170204
» http://doi.org/10.1371/journal.pone.0170204
• 24
Rejnö A, Nasic S, Bjälkefur K, Bertholds E, Jood K. Changes in functional outcome over five years after stroke. Brain Behav. 2019;9(6):e01300. doi: http://doi.org/10.1002/brb3.1300
» http://doi.org/10.1002/brb3.1300
• 25
Magwood GS, Nichols M, Jenkins C, Logan A, Qanungo S, Zigbuo-Wenzler E, et al. Community-based interventions for stroke provided by nurses and community health workers: A Review of the Literature. J Neurosci Nurs. 2020;52(4):152-9. doi: 10.1097/JNN.000000000000512
» https://doi.org/10.1097/JNN.000000000000512

### Edited by

Associate Editor: Maria Lúcia do Carmo Cruz Robazzi

# Publication Dates

• Publication in this collection
03 Sept 2021
• Date of issue
2021


text:
### Top 10 Arxiv Papers Today in Data Structures And Algorithms

##### #1. S-RASTER: Contraction Clustering for Evolving Data Streams
###### Gregor Ulm, Simon Smith, Adrian Nilsson, Emil Gustavsson, Mats Jirstrand
Contraction Clustering (RASTER) is a very fast algorithm for density-based clustering, which requires only a single pass. It can process arbitrary amounts of data in linear time and in constant memory, quickly identifying approximate clusters. It also exhibits good scalability in the presence of multiple CPU cores. Yet, RASTER is limited to batch processing. In contrast, S-RASTER is an adaptation of RASTER to the stream processing paradigm that is able to identify clusters in evolving data streams. This algorithm retains the main benefits of its parent algorithm, i.e. single-pass linear time cost and constant memory requirements for each discrete time step in the sliding window. The sliding window is efficiently pruned, and clustering is still performed in linear time. Like RASTER, S-RASTER trades off an often negligible amount of precision for speed. It is therefore very well suited to real-world scenarios where clustering does not happen continually but only periodically. We describe the algorithm, including a discussion of...
more | pdf | html
None.
###### Tweets
arxiv_cs_LG: S-RASTER: Contraction Clustering for Evolving Data Streams. Gregor Ulm, Simon Smith, Adrian Nilsson, Emil Gustavsson, and Mats Jirstrand https://t.co/9YS6UAvpBt
Memoirs: S-RASTER: Contraction Clustering for Evolving Data Streams. https://t.co/vPjdsXnBbm
None.
None.
###### Other stats
Sample Sizes : None.
Authors: 5
Total Words: 5320
Unqiue Words: 1625

##### #2. Energy consumption in compact integer vectors: A study case
In the field of algorithms and data structures analysis and design, most of the researchers focus only on the space/time trade-off, and little attention has been paid to energy consumption. Moreover, most of the efforts in the field of Green Computing have been devoted to hardware-related issues, being green software in its infancy. Optimizing the usage of computing resources, minimizing power consumption or increasing battery life are some of the goals of this field of research. As an attempt to address the most recent sustainability challenges, we must incorporate the energy consumption as a first-class constraint when designing new compact data structures. Thus, as a preliminary work to reach that goal, we first need to understand the factors that impact on the energy consumption and their relation with compression. In this work, we study the energy consumption required by several integer vector representations. We execute typical operations over datasets of different nature. We can see that, as commonly believed, energy...
more | pdf | html
None.
###### Tweets
powturbo: 🆕🔥🗜"Energy consumption in compact integer vectors: A study case" 🔗 https://t.co/DGLI1NP3Ic #⃣ #DataCompression #IntegerCompression #DataStructures #Algorithms #EnergyConsumption https://t.co/TFoxg7lPb7
None.
None.
###### Other stats
Sample Sizes : None.
Authors: 2
Total Words: 0
Unqiue Words: 0

##### #3. The Power of Factorization Mechanisms in Local and Central Differential Privacy
###### Alexander Edmonds, Aleksandar Nikolov, Jonathan Ullman
We give new characterizations of the sample complexity of answering linear queries (statistical queries) in the local and central models of differential privacy: *In the non-interactive local model, we give the first approximate characterization of the sample complexity. Informally our bounds are tight to within polylogarithmic factors in the number of queries and desired accuracy. Our characterization extends to agnostic learning in the local model. *In the central model, we give a characterization of the sample complexity in the high-accuracy regime that is analogous to that of Nikolov, Talwar, and Zhang (STOC 2013), but is both quantitatively tighter and has a dramatically simpler proof. Our lower bounds apply equally to the empirical and population estimation problems. In both cases, our characterizations show that a particular factorization mechanism is approximately optimal, and the optimal sample complexity is bounded from above and below by well studied factorization norms of a matrix associated with the queries.
more | pdf | html
None.
###### Tweets
ccanonne_: @AcharyaJayadev @thesasho I *knew* this would happen. Just not that soon. https://t.co/u3mFllhr6j The list grows. I'll end up buried under a stack of interesting papers. (Joke aside: this does look great, @thesasho.) https://t.co/XsEZfHKK08
thesasho: After years of talking about DP, Jon Ullman and I finally wrote a paper together https://t.co/WIaThvAFql. Also first paper with Alex Edmonds, supervised by me and Toni Pitassi. We give an essentially optimal algorithm to answer any set of statistical queries in local DP.
Memoirs: The Power of Factorization Mechanisms in Local and Central Differential Privacy. https://t.co/IskrNmyvTo
heghbalz: RT @ccanonne_: @AcharyaJayadev @thesasho I *knew* this would happen. Just not that soon. https://t.co/u3mFllhr6j The list grows. I'll end u…
bipr: RT @thesasho: After years of talking about DP, Jon Ullman and I finally wrote a paper together https://t.co/WIaThvAFql. Also first paper wi…
None.
None.
###### Other stats
Sample Sizes : None.
Authors: 3
Total Words: 0
Unqiue Words: 0

##### #4. Consistent recovery threshold of hidden nearest neighbor graphs
Motivated by applications such as discovering strong ties in social networks and assembling genome subsequences in biology, we study the problem of recovering a hidden $2k$-nearest neighbor (NN) graph in an $n$-vertex complete graph, whose edge weights are independent and distributed according to $P_n$ for edges in the hidden $2k$-NN graph and $Q_n$ otherwise. The special case of Bernoulli distributions corresponds to a variant of the Watts-Strogatz small-world graph. We focus on two types of asymptotic recovery guarantees as $n\to \infty$: (1) exact recovery: all edges are classified correctly with probability tending to one; (2) almost exact recovery: the expected number of misclassified edges is $o(nk)$. We show that the maximum likelihood estimator achieves (1) exact recovery for $2 \le k \le n^{o(1)}$ if $\liminf \frac{2\alpha_n}{\log n}>1$; (2) almost exact recovery for $1 \le k \le o\left( \frac{\log n}{\log \log n} \right)$ if $\liminf \frac{kD(P_n||Q_n)}{\log n}>1$, where $\alpha_n \triangleq -2 \log \int \sqrt{d P_n d... more | pdf | html ###### Figures None. ###### Tweets Memoirs: Consistent recovery threshold of hidden nearest neighbor graphs. https://t.co/dIGIwjgbEv SRoyLee: Consistent recovery threshold of hidden nearest neighbor graphs - https://t.co/0nV3TOALuA ###### Github None. ###### Youtube None. ###### Other stats Sample Sizes : None. Authors: 4 Total Words: 0 Unqiue Words: 0 ###### 1.998 Mikeys ##### #5. Improved Compressed String Dictionaries ###### Nieves R. Brisaboa, Ana Cerdeira-Pena, Guillermo de Bernardo, Gonzalo Navarro We introduce a new family of compressed data structures to efficiently store and query large string dictionaries in main memory. Our main technique is a combination of hierarchical Front-coding with ideas from longest-common-prefix computation in suffix arrays. Our data structures yield relevant space-time tradeoffs in real-world dictionaries. We focus on two domains where string dictionaries are extensively used and efficient compression is required: URL collections, a key element in Web graphs and applications such as Web mining; and collections of URIs and literals, the basic components of RDF datasets. Our experiments show that our data structures achieve better compression than the state-of-the-art alternatives while providing very competitive query times. more | pdf | html ###### Figures None. ###### Tweets powturbo: 🆕🗜"Improved Compressed String Dictionaries" new family of CDS to efficiently store and query large string dictionaries in main memory... 📄https://t.co/Xk8PrxVAok #⃣ #DataCompression #IntegerCompression #InformationRetrieval #DataStructures #Algorithms https://t.co/rXIPsxlnMd okateim: Improved Compressed String Dictionaries https://t.co/YALkWfyOgC ###### Github None. ###### Youtube None. ###### Other stats Sample Sizes : None. Authors: 4 Total Words: 0 Unqiue Words: 0 ###### 1.998 Mikeys ##### #6. Learning-Assisted Competitive Algorithms for Peak-Aware Energy Scheduling ###### Russell Lee, Mohammad H. Hajiesmaili, Jian Li In this paper, we study the peak-aware energy scheduling problem using the competitive framework with machine learning prediction. With the uncertainty of energy demand as the fundamental challenge, the goal is to schedule the energy output of local generation units such that the electricity bill is minimized. While this problem has been tackled using classic competitive design with worst-case guarantee, the goal of this paper is to develop learning-assisted competitive algorithms to improve the performance in a provable manner. We develop two deterministic and randomized algorithms that are provably robust against the poor performance of learning prediction, however, achieve the optimal performance as the error of prediction goes to zero. Extensive experiments using real data traces verify our theoretical observations and show 15.13% improved performance against pure online algorithms. more | pdf | html ###### Figures None. ###### Tweets ###### Github None. ###### Youtube None. ###### Other stats Sample Sizes : None. Authors: 3 Total Words: 10370 Unqiue Words: 2022 ###### 1.998 Mikeys ##### #7. Property Testing of LP-Type Problems ###### Rogers Epstein, Sandeep Silwal Given query access to a set of constraints$S$, we wish to quickly check if some objective function$\varphi$subject to these constraints is at most a given value$k$. We approach this problem using the framework of property testing where our goal is to distinguish the case$\varphi(S) \le k$from the case that at least an$\epsilon$fraction of the constraints in$S$need to be removed for$\varphi(S) \le k$to hold. We restrict our attention to the case where$(S, \varphi)$are LP-Type problems which is a rich family of combinatorial optimization problems with an inherent geometric structure. By utilizing a simple sampling procedure which has been used previously to study these problems, we are able to create property testers for any LP-Type problem whose query complexities are independent of the number of constraints. To the best of our knowledge, this is the first work that connects the area of LP-Type problems and property testing in a systematic way. Among our results is a tight upper bound on the query complexity of... more | pdf | html ###### Figures None. ###### Tweets ###### Github None. ###### Youtube None. ###### Other stats Sample Sizes : None. Authors: 2 Total Words: 0 Unqiue Words: 0 ###### 1.998 Mikeys ##### #8. Optimal Single-Choice Prophet Inequalities from Samples ###### Aviad Rubinstein, Jack Z. Wang, S. Matthew Weinberg We study the single-choice Prophet Inequality problem when the gambler is given access to samples. We show that the optimal competitive ratio of$1/2$can be achieved with a single sample from each distribution. When the distributions are identical, we show that for any constant$\varepsilon > 0$,$O(n)$samples from the distribution suffice to achieve the optimal competitive ratio ($\approx 0.745$) within$(1+\varepsilon)$, resolving an open problem of Correa, D\"utting, Fischer, and Schewior. more | pdf | html ###### Figures None. ###### Tweets DO: Optimal Single-Choice Prophet Inequalities from Samples. https://t.co/bvp49WvhGZ ###### Github None. ###### Youtube None. ###### Other stats Sample Sizes : None. Authors: 3 Total Words: 0 Unqiue Words: 0 ###### 1.998 Mikeys ##### #9. Low-Rank Toeplitz Matrix Estimation via Random Ultra-Sparse Rulers ###### Hannah Lawrence, Jerry Li, Cameron Musco, Christopher Musco We study how to estimate a nearly low-rank Toeplitz covariance matrix$T$from compressed measurements. Recent work of Qiao and Pal addresses this problem by combining sparse rulers (sparse linear arrays) with frequency finding (sparse Fourier transform) algorithms applied to the Vandermonde decomposition of$T$. Analytical bounds on the sample complexity are shown, under the assumption of sufficiently large gaps between the frequencies in this decomposition. In this work, we introduce random ultra-sparse rulers and propose an improved approach based on these objects. Our random rulers effectively apply a random permutation to the frequencies in$T$'s Vandermonde decomposition, letting us avoid frequency gap assumptions and leading to improved sample complexity bounds. In the special case when$T\$ is circulant, we theoretically analyze the performance of our method when combined with sparse Fourier transform algorithms based on random hashing. We also show experimentally that our ultra-sparse rulers give significantly more robust...
more | pdf | html
None.
None.
###### Other stats
Sample Sizes : None.
Authors: 4
Total Words: 5386
Unqiue Words: 1720

##### #10. Outlier-Robust High-Dimensional Sparse Estimation via Iterative Filtering
###### Ilias Diakonikolas, Sushrut Karmalkar, Daniel Kane, Eric Price, Alistair Stewart
We study high-dimensional sparse estimation tasks in a robust setting where a constant fraction of the dataset is adversarially corrupted. Specifically, we focus on the fundamental problems of robust sparse mean estimation and robust sparse PCA. We give the first practically viable robust estimators for these problems. In more detail, our algorithms are sample and computationally efficient and achieve near-optimal robustness guarantees. In contrast to prior provable algorithms which relied on the ellipsoid method, our algorithms use spectral techniques to iteratively remove outliers from the dataset. Our experimental evaluation on synthetic data shows that our algorithms are scalable and significantly outperform a range of previous approaches, nearly matching the best error rate without corruptions.
more | pdf | html
None.
###### Tweets
StatsPapers: Outlier-Robust High-Dimensional Sparse Estimation via Iterative Filtering. https://t.co/6a9KTJxHP1
paulportesi: RT @StatsPapers: Outlier-Robust High-Dimensional Sparse Estimation via Iterative Filtering. https://t.co/6a9KTJxHP1
None.
None.
###### Other stats
Sample Sizes : None.
Authors: 5
Total Words: 15097
Unqiue Words: 3010

Assert is a website where the best academic papers on arXiv (computer science, math, physics), bioRxiv (biology), BITSS (reproducibility), EarthArXiv (earth science), engrXiv (engineering), LawArXiv (law), PsyArXiv (psychology), SocArXiv (social science), and SportRxiv (sport research) bubble to the top each day.

Papers are scored (in real-time) based on how verifiable they are (as determined by their Github repos) and how interesting they are (based on Twitter).

To see top papers, follow us on twitter @assertpub_ (arXiv), @assert_pub (bioRxiv), and @assertpub_dev (everything else).

To see beautiful figures extracted from papers, follow us on Instagram.

Tracking 226,490 papers.

###### Search
Sort results based on if they are interesting or reproducible.
Interesting
Reproducible
Online
###### Stats
Tracking 226,490 papers.


text:
# What is the molar mass of the solute if "4.18 g" of it dissolved in "36.30 g" of benzene (K_f = 5.12^@ "C"cdot"kg/mol") generates a solution with a freezing point of 2.70^@ "C"? The freezing point of benzene is 5.53^@ "C".

Feb 16, 2017

$\text{208.33 g/mol}$

Once you recognize this is freezing point depression, you can use the following equation:

$\boldsymbol{\Delta {T}_{f} = {T}_{f} - {T}_{f}^{\text{*}} = - i {K}_{f} m}$

where:

• ${T}_{f}$ and ${T}_{f}^{\text{*}}$ are the freezing points of the solution and pure solvent, respectively.
• $i$ is the van't Hoff factor. For non-ionic solutes, i.e. nonelectrolytes, $i = 1$, as there is only one "dissociated" particle per dissolved particle.
• ${K}_{f} = {5.12}^{\circ} \text{C"cdot"kg/mol}$ is the freezing point depression constant for benzene, the solvent.
• $m$ is the molality of the solution, i.e. $\text{mols solute"/"kg solvent}$.

As mentioned, benzene is the solvent, because its ${K}_{f}$ is given, and ${K}_{f}$ is only for solvents. Therefore, we should calculate the mols of the solute and kg of benzene.

Simply from the units of molality, we have:

$m = \left(\text{solute mass" xx 1/"molar mass")/("kg solvent}\right)$

$= \left(\text{4.18 g"xx"mol"/"g solute")/("0.03630 kg benzene}\right)$

Therefore, we can solve for the molar mass later, as long as we can calculate the molality.

DeltaT_f = 2.70^@ "C" - 5.53^@ "C" = -(1)(5.12^@ "C"cdot"kg/mol")(m)

$\implies m = \text{0.5527 mols solute"/"kg solvent}$

Now, we can solve for the molar mass.

$\textcolor{b l u e}{\text{Molar mass") = ("4.18 g solute")/("0.5527 mols solute"/cancel"kg solvent" xx 0.03630 cancel"kg solvent}}$

$=$ $\textcolor{b l u e}{\text{208.33 g/mol}}$


text:
# Calculators and complex numbers (Part 14)

In this series of posts, I explore properties of complex numbers that explain some surprising answers to exponential and logarithmic problems using a calculator (see video at the bottom of this post). These posts form the basis for a sequence of lectures given to my future secondary teachers.

Definition. If $z$ is a complex number, then we define

$e^z = \displaystyle \sum_{n=0}^{\infty} \frac{z^n}{n!}$

Even though this isn’t the usual way of defining the exponential function for real numbers, the good news is that one Law of Exponents remains true. (At we saw in an earlier post in this series, we can’t always assume that the usual Laws of Exponents will remain true when we permit the use of complex numbers.)

Theorem. If $z$ and $w$ are complex numbers, then $e^z e^w = e^{z+w}$.

I will formally prove this in the next post. Today, I want to talk about the idea behind the proof. Notice that

$e^z e^w = \displaystyle \left( 1 + z + \frac{z^2}{2!} +\frac{z^3}{3!} + \frac{z^4}{4!} + \dots \right) \left( 1 + w + \frac{w^2}{2!} + \frac{w^3}{3!} + \frac{w^4}{4!} + \dots \right)$

Let’s multiply this out (ugh!), but we’ll only worry about terms where the sum of the exponents of $z$ and $w$ is 4 or less. Here we go…

$e^z e^w = \displaystyle 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \frac{z^4}{4!} + \dots$

$+ \displaystyle w + wz + \frac{wz^2}{2!} + \frac{wz^3}{3!} + \dots$

$+ \displaystyle \frac{w^2}{2!} + \frac{w^2 z}{2!} + \frac{w^2 z^2}{2! \times 2!} + \dots$

$+ \displaystyle \frac{w^3}{3!} + \frac{w^3 z}{3!} + \dots$

$+ \displaystyle \frac{w^4}{4!} + \dots$

Next, we rearrange the terms according to the sum of the exponents. For example, the terms with $z^3$, $w z^2$, $w^2 z$, and $w^3$ are placed together because the sum of the exponents for each of these terms is 3.

$e^z e^w = 1$

$+ z + w$

$\displaystyle + \frac{z^2}{2} + wz + \frac{w^2}{2}$

$\displaystyle + \frac{z^3}{6} + \frac{wz^2}{2} + \frac{w^2 z}{2} + \frac{w^3}{6}$

$\displaystyle + \frac{z^4}{24} + \frac{w z^3}{6} + \frac{w^2 z^2}{4} + \frac{w^3 z}{6} + \frac{w^4}{24} + \dots$

For each line, we obtain a common denominator:

$e^z e^w = 1$

$+ z + w$

$\displaystyle + \frac{z^2 + 2 z w + w^2}{2}$

$\displaystyle + \frac{z^3 + 3 z^2 w + 3 z w^2 + w^3}{6}$

$\displaystyle + \frac{z^4+ 4 z^3 w + 6 z^2 w^2 + 4 z w^3 + w^4}{24} + \dots$

We recognize the familiar entries of Pascal’s triangle in the coefficients of the numerators, and so it appears that

$e^z e^w = 1 + (z+w) + \displaystyle \frac{(z+w)^2}{2!} + \frac{(z+w)^3}{3!} + \frac{(z+w)^4}{4!} + \dots$

If the pattern on the right-hand side holds up for exponents greater than 4, this proves that $e^z e^w = e^{z+w}$.

So that’s the idea of the proof. The formal proof will be presented in the next post.

For completeness, here’s the movie that I use to engage my students when I begin this sequence of lectures.

## 2 thoughts on “Calculators and complex numbers (Part 14)”

This site uses Akismet to reduce spam. Learn how your comment data is processed.