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# Triggerable tough hydrogels for gastric resident dosage forms

## Abstract

Systems capable of residing for prolonged periods of time in the gastric cavity have transformed our ability to diagnose and treat patients. Gastric resident systems for drug delivery, ideally need to be: ingestible, be able to change shape or swell to ensure prolonged gastric residence, have the mechanical integrity to withstand the forces associated with gastrointestinal motility, be triggerable to address any side effects, and be drug loadable and release drug over a prolonged period of time. Materials that have been primarily utilized for these applications have been largely restricted to thermoplastics and thermosets. Here we describe a novel set of materials, triggerable tough hydrogels, meeting all these requirement, supported by evaluation in a large animal model and ultimately demonstrate the potential of triggerable tough hydrogels to serve as prolonged gastric resident drug depots. Triggerable tough hydrogels may be applied in myriad of applications, including bariatric interventions, drug delivery, and tissue engineering.

## Introduction

Devices with the ability to reside in the gastric cavity have been applied for diagnosis1,2,3 and treatment interventions in humans4,5,6,7. The materials used for their fabrication have been largely limited to thermoplastic and thermosets8,9,10. Expanding the repertoire of materials available for system fabrication stands to expand the potential set of applications by enabling greater compatibility with gastrointestinal (GI) tissues, and, moreover, potentially imparting significantly improved safety. When considering future development of novel orally administered systems for prolonged drug release, a key set of elements needs to be considered to ensure safety and potential efficacy of such systems, which are destined to reside in the gastric cavity11,12,13. These include dosage forms in a form factor compatible with ingestion and subsequent shape change to enable gastric residence and prevent passage through the pylorus (the combination of these two elements is analogous to the “ship in the bottle” problem), triggerable elements, which can induce controlled breakage of the dosage form, which can be either administered or are induced by virtue of anatomic location, mechanical, and material properties to ensure stability and integrity in the GI environment, and the ability to be loaded with drug and release drug in a controlled sustained manner.

One potential set of materials with enhanced biocompatibility are hydrogels because of their softness, which can potentially minimize mucosal damage, low polymer content that may reduce the side effects of materials as well as maximize the capacity for dehydration and rehydration in form factors compatible with ingestion and subsequent gastric retention upon swelling, and release drugs in controllable manners14,15,16,17. Conventional hydrogels though generally suffer from being weak and, therefore, can be easily broken by the significant compressive and shearing forces of the GI tract limiting their stability in this environment18. This is exemplified by the application of high viscosity grade cellulose polymers, polyacrylic acid, polyacrylates, polyacrylamides, and polyethers, which have been applied to extending release of drugs on the order of 12–16 h19. Though these systems can provide extension of drug release, they have only enabled the extension of release by hours20. A set of materials with the mechanical properties capable of potentially surviving the gastric environment are tough hydrogels. Though significant advancement has been made in fabricating hydrogels with tougher characteristics, including double-network21,22,23,24,25, topological26,27,28,29,30, and nanocomposite hydrogels31,32,33,34, these gels lack the capacity to be triggered to dissolve in physiological environments, an important property to ensure the maximal safety and removal of such a system that could reside in the gastric cavity for multiple days. Long-term gastric resident systems can be associated with gastric outlet obstruction and/or the development of an allergic reaction could ensue from a gastric drug depot. These complications would require immediate clinical intervention and therefore a system that can be triggered to dissolve and thereby enable passage through the GI tract could have significant advantages over ones without the capacity for triggering where a procedural intervention-like surgery or endoscopy would otherwise be required for removal. Therefore, one essential aspect of safety lies in the ability to trigger the dissolution and controlled fracture of the material to enable rapid transit through the GI tract.

Here we describe a novel set of materials, triggerable tough hydrogels (TTHs), containing up to ~90% water enabling significant shrinkage and swelling to address the “ship in the bottle” challenge. These gels demonstrate stability in normal gastric environments and furthermore can be triggered to dissolve rapidly with biocompatible agents, enabling the development of gastric devices with precisely controlled residence times. We further demonstrate a proof-of-concept gastric resident drug delivery system from these materials in a large animal model. Specifically, the TTHs consist of two types of polymeric networks that are intertwined and separately crosslinked by stimuli-responsive bonds, which can be triggered to dissolve into solution with biocompatible agents on demand. Given the tough and triggerable properties of these hydrogels, we anticipate their application in a broad set of GI biomedical applications.

## Results

### Preparation and characterization of TTHs

TTHs consisting of alginate and polyacrylamide networks are crosslinked by ionic Ca2+ and disulfide bonds, respectively (Fig. 1). Alginate is a linear copolymer comprised of blocks of α-L-guluronic acid, β-D-mannuronic acid, or alternating α-L-guluronic and β-D-mannuronic acids. Divalent Ca2+ cations can crosslink alginate by simultaneously associating with carboxylic groups in the α-L-guluronic acid blocks from different alginate chains, forming an ionically crosslinked network in water21. By contrast, the polyacrylamide network can be formed by aqueous radical polymerization of acrylamide using a bifunctional monomer as the crosslinker. Since alginate and polyacrylamide networks are separately crosslinked, stimuli-responsive ionic and disulfide bonds can be incorporated making the gels susceptible to degradation by biocompatible chelators and reducing agents, TTHs can be de-crosslinked and dissolved into solution accordingly. Alginate and polyacrylamide were selected given their well-recognized biocompatibility and broad biomedical application35.

TTHs were fabricated by a simple one-step method. We dissolved all ingredients needed to form the two networks in deionized water, including sodium alginate and an ionic crosslinker of calcium sulfate for the ionically crosslinked alginate, as well as acrylamide, crosslinking monomer N,N′-bis(acryloyl)cystamine, thermo-initiator of ammonium persulfate, and polymerization accelerator of N,N,N′,N′-tetramethylethylenediamine for the disulfide crosslinked polyacrylamide. The mixture was heated to 50 °C for 1 h and then left in a humid box for 1 day. The unreacted ingredients were purified by continuous extraction with water demonstrating elimination of the acrylamide monomer (Supplementary Fig. 1). Details of the synthesis and characterization of TTHs are described in the Supplementary Information.

We next studied the swelling behavior and variation of mechanical properties of TTHs in simulated gastric fluid (SGF, pH = ~1.2). The TTH swelled progressively (Fig. 2d) and a plateau of volume variation (V t/V 0) of 2.7 ± 0.15 was reached after 6 days of incubation at 37 °C with an accompanying decrease of the tensile properties at rupture. The tensile strength and fracture strain of the TTH decreased to 74.1 ± 6.7 kPa and 12.2 ± 1.1, 62.4 ± 4.9 kPa and 10.1 ± 0.8, as well as 49.2 ± 5.7 kPa and 8.8 ± 0.9 after incubated for 4, 8, and 12 days, respectively (Supplementary Fig. 3a). The swelling also adversely affected the maximum compressive stress of the TTH, which decreased to 2.24 ± 0.24 MPa after 4 days of incubation in SGF (Fig. 2e). After the initial decrease mainly attributed to swelling; however, the maximum compressive stress of the TTH appeared to plateau with further incubation, as confirmed by a small change from 2.21 ± 0.18 to 2.07 ± 0.15 MPa between 8 and 12 days of incubation, respectively. It is worth mentioning that the maximum gastric pressure in the fasted and fed states in humans is known to range from 0.01 to 0.013 MPa36, 37, which is far lower than the maximum compressive stress of the TTH even following incubation periods of up to 12 days, suggesting the potential of these gels to resist gastric compression and achieve relative long-term residence in the gastric cavity. We further measured the dehydration and rehydration of TTHs and found that air-drying effectively dehydrated and shrunk the gel significantly (Supplementary Fig. 3b). Scanning electron microscopy (SEM) images displayed a uniform structure of the dried TTH sample (Supplementary Fig. 4a). As expected, the TTH could not dehydrate into a smaller size by lyophilization and a micro-porous structure was obtained for the lyophilized sample (Supplementary Fig. 4b, c). We next measured the rehydration of TTHs in SGF and found that a completely dehydrated TTH with similar dimensions to a standard 000 capsule swelled to a size greater than the diameter of the resting human pylorus (12.8 ± 7.0 mm)38, 39 within 70 min (Fig. 2f), which is within the 50th percentile for gastric emptying in humans40. In addition, the dehydrated TTH could swell to a size larger than the diameter of pylorus within 15 min in a neutral pH approximating the fed state or patients taking antacids or that can be achieved by co-administration with antacids. The enhanced swelling is attributed to the higher solubility of alginate in neutral pH than in an acidic environment. We found that the adequately rehydrated gel demonstrated a maximum compressive stress of 2.02 ± 0.18 MPa (Supplementary Fig. 5), demonstrating the retention of toughness of TTHs after a cycle of complete dehydration and subsequent rehydration. Alternatively, a TTH-based encapsulation system encasing CaCO3 in an initial form factor of a standard 000 capsule swelled to 27 mm within 30 min in SGF (Supplementary Fig. 6a, b). Similar strategies can be applied for enabling flotation of TTHs (Supplementary Fig. 6c). Given the highly stretchable and tough characteristics, various dosage forms compatible with ingestion and subsequent gastric residence through size exclusion could be developed by using TTHs.

Initial biocompatibility of TTHs was evaluated through in vitro cell toxicity analysis. The gel was incubated in cell culture medium across a wide range of concentrations from 0.2 to 50 mg ml−1 at 37 °C for 24 h. The medium was then tested for its cytotoxicity on multiple cell lines, including HeLa, Caco-2 (C2BBe1 clone), and HT29-MTX-E12 (Supplementary Fig. 7). No significant cytotoxicity was observed for the medium incubated with the gel in any of these cell lines at the end of a 24 h culture period. We next conducted extended cytotoxicity analysis by culturing the TTHs with intestinal stem cells (ISCs) and demonstrate excellent cytocompatibility of the TTHs with mouse Lgr5+ stem cells over the course of 5 days (Supplementary Fig. 8a). Furthermore, we showed that Lgr5+ stem cells could be cultured on and within TTHs and these retained their ability of multilineage differentiation to form organoids (Supplementary Fig. 8b), supporting the biocompatibility and potential application of TTHs serving as a substrate for organoid culture.

### Triggerable properties of TTHs

We next investigated the stimuli-responsiveness of TTHs by using ethylenediaminetetraacetic acid (EDTA) and glutathione (GSH) as triggers of the Ca2+ ion and disulfide crosslinks. Both EDTA and GSH have been previously used in humans as treatments or supplements with oral dosages of up to 6 and 5 g daily, respectively41, 42. In addition, we recognized that both EDTA and GSH have been used as additives in foods43, 44. To measure the triggerable properties and potential boundaries set by EDTA and GSH found in a human diet, TTHs were incubated at 37 °C with a range of concentrations from 20 to 80 mM of EDTA and GSH well above the concentrations found in food for various time intervals, and then evaluated for compressive stress to characterize the dissolution behavior of the gels. Interestingly and supporting the selectivity of the EDTA and GSH combined triggering solution, the TTH could not be dissolved by incubation with EDTA or GSH alone even when incubation times were increased to 24 h (Supplementary Fig. 9). These data support the ability to maintain a network by the remaining crosslinked single network hydrogel and demonstrate that de-crosslinking of both alginate and polyacrylamide networks are essential to dissolve the TTH. The requirement for both EDTA and GSH for triggering supports the likely sustained stability of the TTH in the presence of a normal human diet. The dissolution of the gels was accelerated by triggering with EDTA and GSH simultaneously. As shown in Fig. 3a, the compressive stress of the TTH decreased rapidly from 373 ± 10 to 66.3 ± 5.8, 41.3 ± 3.9, and 16.7 ± 1.0 kPa after 1, 2, and 4 h incubation in 20 mM of EDTA and 20 mM of GSH. When the concentration of EDTA was increased to 40 mM while the GSH was kept constant, the compressive stress of the TTH reduced dramatically to 5.6 ± 0.04 kPa and the gel dissolved after 2 h of incubation. The TTH started to dissolve into a viscous solution only after 1 h incubation with further increases of EDTA to 80 mM (Fig. 3b). In contrast, increases in GSH concentration retarded the dissolution of the TTH (Supplementary Fig. 10), suggesting that the carboxyl group at the C terminus of GSH could disturb the formation of the ionic bond between the Ca2+ and the carboxyl groups in EDTA when excessive GSH was present. Gel permeation chromatography (GPC) of the dissolved TTH demonstrated two peaks with molecular mass of ~120 and ~200 kDa that corresponded to the dissociated alginate and polyacrylamide chains, respectively, supporting the dissolution of the TTH into free polymers (Supplementary Fig. 11). In vitro cell viability assays verified the low cytotoxicity of these dissociated free polymers against HeLa, Caco-2, and HT29 cell lines at the end of a 24 h culture with concentrations up to 5 mg ml−1 (Supplementary Fig. 12).

Having confirmed in vitro the superior stimuli-responsiveness of the gels, we next tested the in vivo dissolution of TTHs by using a Yorkshire pig animal model, which has been previously established for the evaluation of GI resident systems13, 45. Yorkshire pigs weighing 45–55 kg have gastric and intestinal anatomy and dimensions similar to humans46. TTH strips with dimensions 50 mm × 10 mm × 5 mm were introduced endoscopically into the stomach. Pigs were administered a triggering solution consisting of 0.5 L of EDTA (40 mM) and GSH (20 mM) after deployment of the TTH strips. Control samples were deployed into the stomach without the addition of the triggering solution. The TTH strips were retrieved endoscopically after 1 h in the gastric cavity. Strips retrieved from the control pigs remained intact and retained a maximum compressive stress of 1.77 ± 0.15 MPa (Supplementary Fig. 13), whereas the strips from the treated pigs dissolved into viscous solution (Fig. 3c). To further view the in situ dissolution of TTHs in stomach, large TTH sheets, in the shape of an equilateral triangle (side length, 10 mm; thickness, 3 mm) were prepared and labeled with methyl blue. These were triggered in situ with the EDTA/GSH solution and endoscopic videography was used for image capture. Endoscopic video revealed that the TTH sheets were triggered to dissolve within 1 h in the gastric cavities of the treated pigs, whereas the sheets in the control pigs remained intact (Fig. 3d). These results support that TTHs can be triggered to dissolve in vivo with biocompatible agents. We note that long-term (>24 h) gastric resident systems present risks to patients including GI mechanical obstruction and the inability to discontinue a drug in the event of developing an allergic reaction through non-invasive means. The ability to trigger the dissolution of such systems is therefore essential for safe clinical implementation. The need for triggering is further amplified in resource-constrained settings, where healthcare interventions-like endoscopy and surgery may be largely limited and where the inability to remove such systems could manifest in significant morbidity and mortality.

### Gastric retentive drug delivery of TTHs

To evaluate the mechanical integrity of TTHs and their potential application as triggerable biomedical materials in gastric resident systems, we fabricated TTH prototypic gastric resident dosage forms. These were evaluated for their gastric residence and integrity in Yorkshire pigs. To evaluate the gastric retention and in vivo integrity of TTHs, radiopaque capsule-like TTH dosage forms with volumes of 22 ml (diameter, 2.8 cm; total length, 5 cm) were designed and prepared by mixing barium sulfate with the pre-gel solution immediately prior to polymerization. It was noted that the significant load of barium sulfate required for radiographic visualization (20 wt%) manifested in slower swelling characteristics than the TTHs (Supplementary Fig. 14). We therefore used barium sulfate-containing TTHs in their hydrated states, which enabled the retention by virtue of the size of the gel administered and radiographic visualization by virtue of their barium content. Four individual experiments in four different pigs were performed and radiographs were taken approximately every 48–72 h to monitor the integrity of the dosage form, its anatomic location, and any evidence of GI obstruction. Intact prototype TTH systems were observed to achieve gastric residence of 7-9 days (Fig. 4a, b). TTHs remained stable in vitro in SGF (>12 days) though in vivo breakage of TTHs was observed earlier than this likely due to the compressive stress associated with gastric motility, and potential de-crosslinking of alginate network by exchange reactions with monovalent cations in the GI environment47. Meanwhile, the disulfide bonds in polyacrylamide network may be reduced by protein or peptide-associated thiols though low-molecular weight thiols, GSH, and cysteine are only present at a low level or even absent in human gastric fluid48. No intact devices were visualized outside of the gastric cavity, supporting that device breakage first occurred in the stomach enabling their eventual passage out of the stomach. Once device breakage occurred, the resulting fragments were visualized in the intestines without evidence of intestinal obstruction (Supplementary Fig. 15). Throughout the experiments, the animals were found to have normal eating and stooling patterns and did not exhibit any signs of GI obstruction, either clinically or radiographically.

In summary, we present a novel family of TTHs and demonstrate their capacity for significant dehydration and rehydration. We further demonstrate their capacity to be triggered to dissolve with the application of biocompatible triggers. TTHs were evaluated for their stability and mechanical integrity in a large animal model. A potential application in drug delivery was also demonstrated with an extended release system for lumefantrine. Preclinical studies will be required to translate these systems for human application including further safety studies and stress testing in other large animal models. In sum, the TTHs described herein present three important points of novelty from the hydrogel perspective: exceptional mechanical properties: that can withstand in vivo gastric forces and achieve long-term residence in the stomach of a large mammal; remarkable triggerable properties: capable of on-demand dissolution; TTHs can be drug loaded and provide controlled drug release. We believe that this combination of features makes TTHs uniquely attractive for the development of advanced gastric dosage forms for prolonged drug delivery, ingestible electronics, and bariatric applications.

## Methods

### Materials

Acrylamide (A9099, ≥99%), N,N′-bis(acryloyl)cystamine (A4929), ammonium persulfate (A3678, ≥98%), N,N,N′,N′-tetramethylethylenediamine (T9281, 99%), sodium alginate (A2033, medium viscosity), calcium sulfate (C3771, ≥99%), methyl blue (M6900), barium sulfate (11844), L-GSH reduced (≥98%), EDTA (≥99%), DMSO (D8418), sodium bicarbonate (NaHCO3, S5761), calcium carbonate (CaCO3, ≥99%), and phosphate-buffered saline (PBS, pH 7.4) were available commercially from Sigma-Aldrich and used as received unless otherwise noted. Insulin was kindly provided by Novo Nordisk and labeled by Alexa-Fluor® 488. Lumefantrine and rifampicin were purchased from Hangzhou Hysen Pharma Co., Ltd in China. Nanopure water (18 MΩ cm) was acquired by means of a Milli-Q water filtration system, Millipore (St. Charles). SGF (pH ~1.2) was made by dissolving 2 g NaCl and 8.3 ml concentrated HCL in nanopure water and adjusting to 1000 ml.

### Mechanical characterization

The mechanical characterization in tension and compression was performed on an Instron testing machine according to ASTM standards D638 (tension) and D575 (compression). For tensile measurement, specimens were loaded into the grips with a 50 N load cell and the gauge length measured using a digital micrometer. Displacement was applied to the specimen at a rate of 0.15 mm s−1 until samples ruptured. For compression measurement, specimens were placed into a constrained loading compression jig with a 500 N load cell and the gauge length measured using a digital micrometer. Displacement was applied to the specimen at a rate of 0.05 mm s−1 until reaching 95% compressive strain. Force was converted into pressure (F/A) and displacement into strain (ΔL/L).

### High performance liquid chromatography (HPLC)

HPLC measurement was carried out on an Agilent 1260 Infinity HPLC system equipped with a quaternary pump, autosampler, thermostat, control module, and diode array detector as described previously45 with minor modifications. The output signal was monitored and processed using the ChemStation® software. Chromatographic separation was carried out on a 50 mm × 4.6 mm EC-C18 Agilent Poroshell 120 analytical column with 2.7 µm spherical particles, maintained at 40 °C. The optimized mobile phase consisted of acetonitrile, methanol, and buffer (pH 3.5 adjusted with 0.1% formic acid) (72:20:8, v/v) at flow rate of 0.5 ml min−1 over a 10 min run time. The injection volume was 4 µl, and the ultraviolet (UV) detection wavelength of 254 nm was selected.

### Liquid chromatography tandem-mass spectrometry

UPLC separation was conducted on a Waters UPLC aligned with a Waters Xevo-TQ-SMS mass spectrometer (Waters Ltd., UK) as described previously45 with minor protocol modifications as follows. MassLynx 4.1 software was used for data acquisition and analysis. Liquid chromatography separation was performed on an Acquity UPLC CSH C18 (50 × 2.1 mm, 1.7 µm particle size) at 50 °C. The mobile phase consisted of acetonitrile, 0.1% formic acid, and 10 mM ammonium formate was flowed at a rate of 0.6 ml min−1 using a time and solvent gradient composition. The initial gradient (100%) was followed by a linear gradient (20%) over 0.25 min. Over the next 1.25 min, the gradient was brought to 0% and held for 0.5 min and finally brought back to the initial gradient of 100% over 0.25 min and held until the end of the run for column equilibration. The total run time was 4 min and sample injection volume was 2.5 µl. The mass spectrometer was operated in the multiple reaction-monitoring mode. Sample introduction and ionization was ESI in the positive ion mode. Stock solutions of lumefantrine and an internal standard artemisinin were prepared in methanol at a concentration of 500 µg ml−1. A ten-point calibration curve was prepared ranging from 2.5 to 2500 ng ml−1. Quality-control samples were prepared in a similar procedure using an independent stock solution at three concentrations (2.5, 25,and 250 ng ml−1). A measure of 200 µl of internal standard 250 ng ml−1 was added to 100 µl of sample solution to cause precipitation. Samples were vortexed and sonicated for 10 min and then placed in a centrifuge for 10 min. A measure of 200 µl of solution was pipetted into a 96-well plate containing 200 µl of water. Finally, 2.5 µl was injected into the Ultra Performance Liquid Chromatography-electrospray ionization-MS system for analysis.

### SEM

Surface morphology of the dehydrated gels was observed using the JEOL 5600LV SEM. For visualization under SEM, samples were fixed to aluminum stubs with double-sided adhesive carbon conductive tape and subsequently sputter coated with carbon using a Hummer 6.2 Sputter System.

### GPC

Aqueous GPC was conducted on a Viscotek system (Malvern) equipped with an isocratic pump Viscotek VE 1122 solvent delivery system, TDA 305 triple detector array, and three TSK Gel GMPWxL column with guard column. The system was equilibrated at 30 °C in pre-filtered water containing 0.05 M NaNO3 with the flow rate set to 1 ml min−1. Polymer solutions were prepared at a concentration of about 0.5-5 mg ml−1 and an injection volume of 200 μl was used. Data collection and analysis were performed with ChemStation for LC (Agilent) and OmniSEC v. 4,6,1,354 software (Malvern). The system was calibrated with poly(ethylene oxide) standards (Sigma) ranging from 400 to 511,000 Da (M p).

### Purification of TTHs

To measure the unreacted ingredients in TTHs, the resulted gel was cut into 1–2 mm pieces and sonicated in 10 volumes of water for 30 min. The mixture was further incubated at 37 °C for 24 h on a shaker plate at 250 r.p.m. After the addition of a certain volume of acetonitrile, the mixture was centrifuged and the supernate was analyzed by HPLC. To purify the TTH, the obtained gel was extensively extracted with 4 × 1000 ml water for 24 h. The same procedure described above was performed to measure the unreacted ingredients in the purified TTH.

### Swelling and stability of TTHs in SGF

The swelling and stability of TTHs were measured by incubating TTH samples in SGF at 37 °C and subsequent measuring the volume as well as the maximum compressive stress. Typically, the cylindrical TTH samples (diameter, 6.2 mm; length, 12 mm) were prepared by carrying out the gelation reaction in a 3.5 ml VWR glass vial. The obtained gels were submerged in 50 ml GSH in a Corning CentriStarTM tube and then incubated at 37 °C on a shaker plate at 250 r.p.m. After predetermined time intervals, the size of the samples was measured by using a digital micrometer and compared with initial volumes. Meanwhile, the TTH samples were also subjected to compression measurement. Three replicates were conducted for each TTH sample.

### Dehydration and rehydration of TTHs

The dehydration of TTHs was measured by incubating TTH samples in air at 37 °C. Typically, the cylindrical TTH samples (diameter, 6.2 mm; length, 12 mm) were placed in the oven set at 37 °C and the size of the samples after predetermined incubation intervals was measured by using a digital micrometer and compared with initial volumes. For rehydration measurement, the dehydrated gel samples were submerged in 50 ml SGF in a Corning CentriStarTM tube and incubated at 37 °C on a shaker plate at 250 r.p.m. After different time intervals, the size of the samples was measured and compared with initial volumes. Three replicates were conducted for each TTH sample. In a control experiment, TTH samples were frozen at −20 °C and subsequently dried by lyophilization.

### Dissolution of TTHs with triggers

The dissolution of TTHs was studied by using EDTA and GSH as triggers. Typically, the TTH were cut into 1 cm3-sized cubes and submerged in 10 ml PBS (pH 7.4) containing EDTA and GSH with a range of concentrations from 20 to 80 mM in a 20 ml VWR glass vial. Three replicates for each time point and condition were incubated at 37 °C on a shaker plate at 250 r.p.m. At each time point, the TTH cubes were subjected to compression measurement. The TTH cubes incubated in either 20 mM of EDTA or GSH were used as controls. To demonstrate the complete dissolution of TTHs into free polymer chains, the triggered solutions were filtered by a 0.2 µm filter and subsequently injected into GPC. For cytotoxicity assay of the dissociated polymers, the triggered solutions were transferred to dialysis tubes (MWCO, 10 kDa), then dialyzed against pure water for 3 days to remove EDTA and GSH, and finally dried by lyophilization.

### Permeability measurement

The measurement of permeability of TTHs was carried out on a Franz diffusion cell using a TTH membrane (thickness, 3 mm). 2 ml PBS (pH 7.4) containing 1 mg ml−1 DMSO, rifampicin, or insulin was added into the donor compartment of the cell and 12 ml fresh PBS was placed in the acceptor compartment of the cell. At each time point, 0.4 ml was sampled from the acceptor compartment and 0.4 ml fresh PBS was supplemented through the sampling port of the cell. The concentration of DMSO and rifampicin of the samples was recorded on a Perkin-Elmer Lambda ultraviolet-visible spectrometer, and the UV absorbance calibration curve of DMSO in a range from 6.25 to 100 µg ml−1 or rifampicin in a range from 1.56 to 100 µg ml−1 with a correlation coefficient >99.9% was used to determine the concentration. The content of Alexa Fluor 488-labeled insulin was measured on an Infinite M200Pro (Tecan) reader (excitation, 490 nm; emission, 540 nm).

### In vitro drug release

Individual 50 mg TTH cubes with lumefantrine content of 1, 5, or 10 wt% were used for long-term release studies. Typically, the TTH cubes were submerged in 2 ml SGF in a 15 ml VWR centrifuge tube and then incubated at 37 °C on a shaker plate at 250 r.p.m. At each time point, the release medium was replaced by 2 ml fresh SGF and then frozen at −80 °C until analysis. The release study was carried out for up to 12 days and the total drug release was measured by HPLC using a linear standard curve of lumefantrine with a range of concentration from 0.005 to 50 µg ml−1 and a correlation coefficient >99.9%.

### Cytotoxicity assay

The cytotoxicity assay of the dissociated polymers was conducted by adding the polymers directly into the culture medium with a range of concentrations from 0.02 to 5 mg ml−1. For the TTH, the gel was incubated in the culture medium with a range of dosage from 0.2 to 50 mg ml−1 at 37 °C for 24 h. The obtained medium was then tested for its toxicity toward cells. Cell lines were purchased from ATCC and Public Health England for these experiments. To avoid cross contamination, expanded cells were stored in individual containers. Regular mycoplasma evaluations were performed of the cell culture environment and the cell lines to ensure the absence of mycoplasma contamination. Cytotoxicity was tested on HeLa, C2BBe1 (ATCC), and HT29-MTX-E12 cells (Public Health England) by seeding them each in a 96-well plate at a density of 10,000 cells per well as described previously13 with minor modifications. HeLa and HT29-MTX-E12 cells were cultured in 100 µl Dulbecco’s modified Eagle medium (DMEM) (Life Technologies) containing 1% non-essential amino acids (Life Technologies), 10% fetal bovine serum (Life Technologies), and 1% penicillin-streptomycin solution (Life Technologies) per well. C2BBe1 cells were cultured in the same medium, but the 1% non-essential amino acids were replaced with 1% human insulin-transferrin-selenium (Life Technologies). Cells were kept in culture for 3 days before replacing the medium with 100 μl of the pre-prepared solutions as described above. After 24 h, these solutions were replaced with 100 μl untreated media and cytotoxicity was quantified by adding 10 µl alamarBlue reagent (Life Technologies) to each well. The contents were mixed and then allowed to incubate at 37 °C for 1 h. Fluorescence was recorded on an Infinite® M200Pro (Tecan) with excitation at 560 nm and emission at 590 nm. A positive control was provided by treating cells with 70% ethanol for 1 hour prior to the cytotoxicity assay experiment. Cells that were not subjected to any polymer-treated media provided a negative control. Cell viability was calculated by the following equation: cell viability (%) = 100 × [Absorbance(sample)−Absorbance(positive control)]/[Absorbance(negative control)−Absorbance(positive control)].

### Stem cell culture

Cell culture media (Advanced DMEM/F12 with N2, B27, and N-acetylcysteine) containing growth factors (EGF, Noggin, and R-Spondin 1) and small molecules (CHIR99021 and VPA) were used for stem cell culture. All animal experiments were performed in accordance with protocols approved by the Committee on Animal Care at MIT. Single mouse Lgr5-GFP ISCs were isolated from Lgr5-EGFP-IRES-CreERT2 mice (Jackson Labs) as described previously50. The isolated single Lgr5-GFP stem cells were cultured in Matrigel for 2 days to form stem cell colonies before use. To evaluate the cytobiocompatibility of TTHs against stem cells, the TTH dishes (thickness: 1 mm, diameter: 8 mm) were cultured directly with Lgr5-GFP stem cell colonies in a 24-well plate for 5 days. To test the ability of TTHs serving as a substrate for organoid culture, Lgr5-GFP stem cell colonies were mixed with Matrigel and then placed on the lyophilized TTH dishes in a 24-well plate. The plate was either placed in a 37 °C incubator directly (for cell culture on the TTHs), or further incubated for 30 min on ice and subsequently centrifuged at 300 × g for 2 min before placed in a 37 °C incubator (for cell culture within the TTHs). Cells were cultured in stem cell culture media for 3 days before switching to organoid culture media (by removing CHIR99021 and VPA to permit spontaneous differentiation of the stem cells) and further cultured for another 4 days.

### Modeling pharmacokinetic data

To determine the elimination rate constant and half-life of the free lumefantrine, a one-compartment oral absorption model was fit to the pharmacokinetic data. The model is shown in Supplementary Fig. 19a. Using this model, an equation was derived to describe the plasma concentration time profile.

$${C_{\rm{p}}} = \frac{{{\rm{Dose }}\,{k_{\rm{a}}}}}{{V\left( {{k_{\rm{a}}} - {k_{\rm{e}}}} \right)}}\left[ {{{\rm e}^{ - {k_{\rm{e}}}t}} - {{\rm e}^{ - {k_{\rm{a}}}t}}} \right]$$
(1)

To describe the pharmacokinetic profile obtained from the lumefantrine-loaded TTH device, a pharmacokinetic model described in Supplementary Fig. 19b was used. An equation was obtained to describe the plasma concentration.

$${C_{\rm{p}}} = \frac{{{\rm{Dose}}\;{k_{{\rm{rel}}}}{k_{\rm{a}}}}}{V}\left[ {\frac{{{{\rm e}^{ - {k_{{\rm{rel}}}}t}}}}{{\left( {{k_{\rm{a}}} - {k_{{\rm{rel}}}}} \right)\left( {{k_{\rm{e}}} - {k_{{\rm{rel}}}}} \right)}} - \left( {\frac{1}{{{k_{\rm{a}}} - {k_{\rm{e}}}}}} \right)\left( {\frac{{{{\rm e}^{ - {k_{\rm{e}}}t}}}}{{{k_{{\rm{rel}}}} - {k_{\rm{e}}}}} - \frac{{{{\rm e}^{ - {k_{\rm{a}}}t}}}}{{{k_{{\rm{rel}}}} - {k_{\rm{a}}}}}} \right)} \right]$$
(2)

### Data availability

The data supporting the findings of this study are available within the article and its Supplementary Information files and from the corresponding authors on reasonable request.

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## Acknowledgements

This work was funded in part by the Bill and Melinda Gates Foundation Grants OPP1096734 and OPP1139927 and the NIH Grant EB000244. The paper was partly sponsored by the Alexander von Humboldt Foundation under the auspices of the Max Planck Research Award to R.L. funded by the Federal Ministry of Education and Research.

## Author information

Authors

### Contributions

J.L. designed the material and experiments. R.L. and G.T. supervised the research. J.L. prepared the material and the device. J.L., Y.P., S.Z., C.C., X.Y., L.B., J.L., Y.-A.L.L., H.M., S.S., A.R.K., T.E., J.R.,R.L., and G.T. characterized the material and analyzed the data. J.L., R.L., and G.T. wrote the paper. All authors discussed the progress of research and reviewed the manuscript.

### Corresponding authors

Correspondence to Robert Langer or Giovanni Traverso.

## Ethics declarations

### Competing interests

J.L., R.L., and G.T. are co-inventors on a provisional patent application number 62/419650 describing the TTHs, which was filed on 9 November 2016. The remaining authors declare no competing financial interests.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Liu, J., Pang, Y., Zhang, S. et al. Triggerable tough hydrogels for gastric resident dosage forms. Nat Commun 8, 124 (2017). https://doi.org/10.1038/s41467-017-00144-z

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• ### Temperature/near-infrared light-responsive conductive hydrogels for controlled drug release and real-time monitoring

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# Enhancement of Photon Absorption on$${\text{Ba}}_{x}{\text{Sr}}_{1-x}{\text{TiO}}_{3}$$Thin-Film Semiconductor Using Photonic Crystal

International Journal of Optics

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### Abstract

Enhancement of photon absorption on barium strontium titanate ( $Ba x Sr 1 - x TiO 3$) thin-film semiconductor for mole fraction $x = 0.25$, 0.35, 0.45, and 0.55 using one-dimensional photonic crystal with defect was investigated experimentally. The thin film was grown on transparent conductive oxide (TCO) substrate using chemical solution deposition method and annealed at 500°C for 15 hours with increasing rate of 1.6°C/min. From optical characterization in visible spectrum it was found that the average absorption percentages are 92.04%, 83.55%, 91.16%, and 80.12%, respectively. The BST thin film with embedded photonic crystal exhibited a relatively significant enhancement on photon absorption, with increasing value of 3.96%, 7.07%, 3.04%, and 13.33% for the respective mole fraction and demonstrating absorbance characteristic with flat feature. In addition, we also discuss the thin-film properties of attenuation constant and electrical conductivity.

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### Improving thin-film crystalline silicon solar cell efficiencies with photonic crystals.

(2007)
Most photovoltaic (solar) cells are made from crystalline silicon (c-Si), which has an indirect band gap. This gives rise to weak absorption of one-third of usable solar photons. Therefore, improved light trapping schemes are needed, particularly for c-Si thin film solar cells. Here, a photonic crystal-based light-trapping approach is analyzed and compared to previous approaches. For a solar cell made of a 2 mum thin film of c-Si and a 6 bilayer distributed Bragg reflector (DBR) in the back, power generation can be enhanced by a relative amount of 24.0% by adding a 1D grating, 26.3% by replacing the DBR with a six-period triangular photonic crystal made of air holes in silicon, 31.3% by a DBR plus 2D grating, and 26.5% by replacing it with an eight-period inverse opal photonic crystal.
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### Crystalline Si thin-film solar cells: a review

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# Chapter 14 Hypothesis Testing: One Sample

library(HannayIntroStats)

## 14.1 Introduction and Warning

We now turn to the art of testing specific hypotheses using data. This is called Hypothesis testing. Unfortunately, hypothesis testing is probably the most abused concept in statistics. It can be very subtle and should only be used when the question being considered fits snugly into the hypothesis testing framework. We will see that very often a confidence interval is a better choice and conveys more information than is contained in a statistical hypothesis test.

## 14.2 A Starting Example

Lets say that we are investigating whether a large chemical company has been leaking toxic chemicals into the water supply of a town aptly called Chemical City. Let’s say that the toxic chemical is known to stunt the growth of children. Therefore, we collect a random sample of heights of 3rd Grade children in Chemical City by measuring the heights of 42 children. Lets load it into R for analysis.

data(chemical_children)
hist(chemical_children$Heights, main='Heights of Children in Chemical City N=42', ylab='Number of Children', xlab='Height in ft', col='lightblue') We can estimate the population mean $$\mu$$ for all the 3rd grade children in Chemical City using the sample mean: mean(chemical_children$Heights)
## [1] 3.679128

Suppose that we also know from a large study that the mean height for healthy 3rd grade children is $$\mu=3.85$$ feet. The question we can now ask is whether we have collected sufficient evidence that the growth of children in Chemical City has been stunted? Could the average height of the children been less than the national average in our sample by random chance?

We can now form what is called the null hypothesis $$H_0$$ we wish to test.

• $$H_0$$ is the hypothesis that the mean height of children in Chemical city is equal to or greater than the national average $$E[\bar{X}]=\mu \geq 3.85$$

The null hypothesis is often the boring hypothesis which doesn’t change the status quo. The opposite of the null hypothesis is called the alternative hypothesis. In this example our alternative hypothesis would be:

• $$H_a$$ is the hypothesis that the mean height of children in Chemical city is less than the national average $$E[\bar{X}]< \mu$$.

As usual, our first approach will be based on simulations. We can re-sample of height data to get an idea of how variable our sample mean $$\bar{X}$$ will be for children in Chemical City.

num.sim<-100000
many.studies<-replicate(num.sim, mean(sample(chemical_children$Heights, size=42, replace=TRUE))) ##generate a bunch of random sample means length(many.studies[many.studies>=3.85])/num.sim ##find the fraction of random samples which give a value greater than the national average ## [1] 0.01296 Therefore, we can say that their is a small (about 1%) chance that our sample produced a smaller result than the national average by random chance. ## 14.3 The t.test command: Hypothesis Tests for the Population Mean $$\mu$$ For large enough $$n \geq 30$$ samples sizes (the same conditions for using the t.test command for confidence intervals) we can also do hypothesis testing using the t.test command for a hypothesis concerning the population mean. For example, here is the command for doing the Chemical City problem. t.test(chemical_children$Heights, mu=3.85, alternative="less")
##
##  One Sample t-test
##
## data:  chemical_childrenHeights ## t = -2.1814, df = 41, p-value = 0.01747 ## alternative hypothesis: true mean is less than 3.85 ## 95 percent confidence interval: ## -Inf 3.810953 ## sample estimates: ## mean of x ## 3.679128 The first parameter is the data set, the second is the null hypothesis value for the mean and the third option tells R that our alternative hypothesis is that the mean is less than 3.85. We will see examples of the other options for this. Notice that R will give you a sentence in the output of the t.test spelling out what the alternative hypothesis is. Also, notice the line which gives us the p.value. The p-value may be used to determine how well the data align with results expected when the null hypothesis is true. It is a measurement of how compatible the data are with the null hypothesis. How likely is the effect observed in the sample data if the null hypothesis is true. The p value gives the probability of observing our data as extreme (or more extreme) assuming the null hypothesis is true. • High p-values The data are likely when the null is true • Low p-values The data are unlikely under the null hypothesis. A low p-value implies we can reject the null hypothesis. In this case we can see that the p.value is pretty small $$p\approx0.01$$ so we have strong evidence against the null hypothesis in our data. Therefore, we might reject the hypothesis that the children of Chemical City have average or greater heights. Notice that we haven’t “proved” the alternative hypothesis conclusively, rather we have found it is unlikely that the null hypothesis holds in light of the evidence present in the sample. The one sample t test is making use of the central limit theorem under the hood. Therefore it comes with the same limitations as the CLT. • Only works for the hypothesis tests concerning the mean • Need sample sizes of at least 30 to be safe • If we apply this with smaller samples, we are assuming that the underlying population distribution is roughly mound shaped. Generally, we have no way to verify this. The statements above might remind you of a conditional probability statement. In fact we can think of the p value as the conditional probability $P(D|H_0),$ that is the probability of observing the data given that the null hypothesis is true. Recall that generally $$P(A|B)\neq P(B|A)$$. This error leads to a common misinterpretation of p values that they give the reverse conditional probability $$P(H_0|D)$$. It is not correct to say that p values give the probability the null hypothesis is true given the data. This is really a more useful probability to know in many ways. However, that is NOT what a p value gives us. To reverse conditional probabilities we need Bayes Rule. ## 14.4 Theory of Hypothesis Testing We have already seen the main ingredients of statistical hypothesis testing. However, here are the basic steps for single sample hypothesis tests for the population mean $$\mu$$: 1. Select the alternative hypothesis as that which the sampling experiment is intended to establish. The alternative hypothesis will take one of the following forms: • One-tailed, upper tailed (e.g $$H_a: \mu > 2400$$) • One tailed, lower tailed (e.g $$H_a: \mu < 2400$$) • Two-tailed (e.g $$H_a \neq 2400$$) 2. Select the null hypothesis $$H_0$$ as the status quo. It will be the opposite of the alternative hypothesis, although if the alternative is one-tailed we can fix the value at the closest value to the alternative hypothesis. 3. Run the test in R using the t.test command. t.test(data, mu=null value, alternative=two.sided, less, greater) 1. Interpret the results using the p.value. If p is small we may reject the null hypothesis, if it is large we say that we retain the null hypothesis, or fail to reject the null hypothesis. More on the reasons for this jargon later, although I will say that using the correct jargon is very important to avoiding misinterpretations of hypothesis testing results. For each of the following situations form the null and alternative hypothesis and give the options for the command in R. Exercise 14.1 In a high school they have tested out a new teaching technique on a class. The students scored an average of an 82 on a standardized test, where the national average is a 75. The school wants to know if the new teaching technique has improved the test scores. Exercise 14.2 Lets say we have collected data on tadpole sizes in one pond in Chemical city. If the average tadpole size is 1cm we would like to know if these chemical city tadpoles are abnormally small. Exercise 14.3 The Lays potato chip factory has added a new machine which makes bags of potato chips. If Lays machines typically put an average of 17 chips into each bag, we want to know if the new machine is performing like the typical potato chip machines. If you are conducting a hypothesis test and get a p value which is near 1 this might be a clue that you have your alternative backwards. ## 14.5 Under the Hood (t tests) Suppose we have designed a questionnaire system (SMARTY PANTS) to select people with higher than average IQs. The IQ test by definition gives the average person a score of 100. To evaluate this system we take a random sample of 50 people which SMARTY PANTS tells us have a higher than average IQ and have them complete an actual IQ test. We find that the mean IQ score of our SMARTY PANTS selected people is $$m=103$$ with a sample standard deviation of $$s=15$$. This sample mean (m) is certainly greater than 100, but we know this could have occurred by chance. Perhaps we were just lucky in our samples? To evaluate this we want to conduct a statistical hypothesis test with: • $$H_0$$: $$\mu \leq 100$$ The mean of those selected people by SMARTY pants in less than or equal to 100 • $$H_a$$: $$\mu > 100$$ The mean of those selected people by SMARTY is greater than 100. To evaluate this we assume that the null hypothesis is true. As we specified above the null hypothesis here owns all values less than or equal to 100 ( $$H_0:\mu \leq 100$$). The closest value in the null hypothesis range to our sample mean (m=103) is 100. So let’s assume that the population (true) mean of our SMARTY PANTS sample is 100. Given this assumption we now want to calculate the probability that we would collect a sample of 50 people with a sample mean greater than or equal to what we actually observed $$m \geq 103$$? Well, we know from our study of sampling that the sampling distribution for the sample mean $$\bar{x}$$ will be approximately normally distributed (for samples with at least 30) with a mean=$$\mu$$ and a standard deviation given by the standard error $$\sigma_{\bar{x}}=s/\sqrt{N}$$. Notice this statement is using the central limit theorem. Figure 14.2 shows the sampling distribution assuming the null hypothesis is true. Recall, we can compute the Z score for normally distributed values to get an idea of how extreme they are. $Z=\frac{D-\mu}{\sigma}=\frac{m-\mu}{\frac{s}{\sqrt{N}}}$ se=15/sqrt(50) Zval=(103-100)/se Zval ## [1] 1.414214 This tells us that the results we observed in our data are 1.4142136 standard deviations above the mean. Thus, the data we observed is not particularly extreme under the null hypothesis. We can find the probability we observe a Z score greater than or equal to 0.2 using pnorm: 1-pnorm(Zval) ## [1] 0.0786496 This is in fact the p value for this statistical hypothesis test. The Z value is called the test statistic in this case. To be more precise we should really use the student t distribution instead of the assuming a normal distribution for our samples. Recall, we use the student t distribution when we have used the data to estimate both the sample mean and the population standard deviation. This is almost always the case for real data sets. Using the student t distribution in R is pretty easy we just switch pnorm for pt and specify the second parameter for a student t distribution (called the degree of freedom). The degrees of freedom for our t distribution is just the number of samples (100) minus 1. 1-pt(Zval, df=99) ## [1] 0.08021821 Notice, this is essentially the same thing as what we found with the normal approximation above. This is because the student t distribution becomes closer and closer to a normal distribution for large sample sizes. In summary when we use a t.test command in R. This is what R computes for us: 1. The test statistic (t): $t=\frac{m-\mu}{\frac{s}{\sqrt{N}}}$ • $$m$$ is the sample mean • $$\mu$$ is the mean under the null hypothesis • $$s$$ is the sample standard deviation • $$N$$ is the sample size. 2. The p value using the alternative keyword: • Greater: 1-pt(t, df=N-1) • Less: pt(t, df=N-1) • Two-tailed: 2*pt(abs(t), df=N-1) where abs is the absolute value of our t statistic. ## 14.6 Errors in Hypothesis Testing In general two things can go wrong when we use statistical hypothesis testing (Fig. 14.3). 1. Type I Error (False Positive): occurs when the researcher rejects the null hypothesis in favor of the alternative hypothesis, while in fact the null hypothesis is true. The probability of committing a type I error is denoted by $$\alpha$$. 2. Type II Error (False Negative): occurs when the researcher retains the null hypothesis when in fact the null hypothesis is false. The probability of commuting a type II error in denoted by $$\beta$$. An analogy that some people find helpful in understanding the two types of error is to consider a defendant in a trial. Court Analogy for Hypothesis Tests: • $$H_0:$$ The defendant is innocent • $$H_a:$$ The defendant is guilty • Type I Error: Convict an innocent person • Type II Error: Let a guilty person go free Notice that we could always set one of these error rates to zero if we wanted. For example, if I just convict everyone that comes to trial then my Type II error would be zero! I cannot let a guilty person go free if I convict everyone. On the other hand I can set my Type I error rate to zero by acquitting every case. This might seem silly but it illustrates the point that in any real scenario (with uncertainity involved) both the Type I and Type II error rates will be non zero. ### 14.6.1 Statistical Significance ($$\alpha$$) We can control the $$\alpha$$ or false positive rate of our statistical test by setting a significance level before we test the hypothesis (really this should be set before we even collect the data). Typically we use $$\alpha=0.05$$ or $$\alpha=0.01$$ which mean that we have a 5% chance or 1% chance respectively of rejecting the null hypothesis incorrectly. When the statistical test is performed then we reject the null hypothesis only if the p-value produced is less than the significance level used. To understand this lets perform some simulations. The below code runs a t test for the null hypothesis $$\mu=1.0$$ for a random sample from a normal distribution which has a mean of 1.0. t.test(rnorm(100, mean=1.0, sd=2.0), mu=1.0, alternative = "two.sided")p.value
## [1] 0.7712665

In this artificial case we know the null hypothesis is true, but due to sampling error we may occasionally get an p value which indicates that the null hypothesis should be rejected. Notice that the p.value which comes out changes every time we collect a sample and run our test.

t.test(rnorm(100, mean=1.0, sd=2.0), mu=1.0, alternative = "two.sided")$p.value ## [1] 0.2429438 This is a key-point the p value depends on the data we collect, which the result of random sampling, therefore p values are random variables themselves! By setting our significance level in advance to $$\alpha=0.05$$ we say that we will reject the null hypothesis if and only if $$p < 0.05$$. Lets run our simulation thousands of times to see how many times we commit a Type I error at this significance level. many.p.vals<-replicate(50000, t.test(rnorm(100, mean=1.0, sd=2.0), mu=1.0, alternative = "two.sided")$p.value)
sum(many.p.vals<0.05)/50000
## [1] 0.05008

We can see that we get about what we expect from this simulation. We incorrectly reject the null hypothesis about 5% of the time as expected. If we were to compute the p value of the test, and then decide to set our significance level we can end up with a much higher false positive rate. This is because very often we are searching for evidence to reject the null hypothesis.

If we conduct enough experiments even when their is no effect whatsoever eventually we will find convincing statistical evidence against the null hypothesis.

For example, if we conduct 100 experiments and only report the smallest p value we find then we can find an effect:

many.pv<-replicate(100, t.test(rnorm(100, mean=1.0, sd=2.0), mu=1.0, alternative = "two.sided")$p.value) min(many.pv) ## [1] 0.00678534 By conducting hypothesis tests again and again until we get the result we want (rejecting the null hypothesis) we can end up with a much higher Type I error rate then $$\alpha$$. This fallacy (called p-hacking) is behind many of the absurd statistics you will see online. Nutrional studies are particurily prone to this type of error. This is why you may have heard that (insert common food) causes cancer and only a few weeks later hear that it cures cancer. Figure 14.4 shows some results on this. ### 14.6.2 Type II Error The type II (False Negative) error rate for statistical hypothesis tests is much more difficult to control than the Type I error rate. However, we can get an idea of this rate by doing some simulations where we know the alternative hypothesis is true. type2.er<-replicate(5000, t.test(rnorm(30,mean=1.1, sd=2.0), mu=1.0, alternative = "greater")$p.value)
sum(type2.er>0.05)/5000 ##cases where the retain the null hypothesis at alpha=0.05, even though it is false
## [1] 0.916

Notice how large this $$\beta$$ rate is! In the general $$\beta$$ will usually increase as we decrease $$\alpha$$.

Generally as we attempt to lower the Type I error rate in hypothesis testing this will result in an increase in the Type II error rate.
type2.er<-replicate(5000, t.test(rnorm(30,mean=1.1, sd=2.0), mu=1.0, alternative = "greater")$p.value) sum(type2.er>0.01)/5000 ##cases where the retain the null hypothesis at alpha=0.01, even though it is false ## [1] 0.9798 That is if we risk more False Negatives if we make our criterion for rejecting the null hypothesis very stringent. In the court analogy this is equivalent to saying by making the burden of proof very high we can control for the rate of false convictions, but making this criteria overly stringent will lead to an increase in the false acquittals. The best way to decrease both $$\alpha$$ and $$\beta$$ is to shed more light on the true nature of the population by increasing the sample size. We can control the false positive rate by choosing $$\alpha$$ small, and decrease $$\beta$$ by increasing the sample size. type2.er<-replicate(5000, t.test(rnorm(3000,mean=1.1, sd=2.0), mu=1.0, alternative = "greater")$p.value) ##larger sample size
sum(type2.er>0.05)/5000 ##cases where the retain the null hypothesis at alpha=0.01, even though it is false
## [1] 0.1292

We can see that by increasing the sample size to large samples we can get a more reasonable Type II (False Negative) error rate.

We may lower the chances of a type II error by collecting more data. The type I error rate is controlled by us in choosing an appropriate $$\alpha$$.

### 14.6.3 Practical Significance versus Statistical Significance

It is also important to note that just because we find the differences are statistically significant it doesn’t mean that they are of any practical significance. For example, suppose we wanted to examine whether an online statistics tutor increased the mean grade point average of the students. If the mean GPA before the tutoring was 3.50 we could do a upper tailed significance test to see if the GPA increased.

• $$H_0: \mu=3.5$$

• $$H_a: \mu > 3.5$$

Lets simulate this process but we will set the mean GPA after the tutoring to $$3.51$$.

gpa.after<-rnorm(10000, mean=3.51, sd=0.2)
t.test(gpa.after, mu=3.50, alternative='greater')
##
##  One Sample t-test
##
## data:  gpa.after
## t = 4.9872, df = 9999, p-value = 3.114e-07
## alternative hypothesis: true mean is greater than 3.5
## 95 percent confidence interval:
##  3.506674      Inf
## sample estimates:
## mean of x
##  3.509959

We can see that the difference in GPAs before and after the tutoring is highly, highly significant with a tiny p value. However, if we payed 500 dollars for the tutoring a difference in 0.01 in the final GPA is of no practical significance. This means we need to pay attention to the actual differences rather than just whether the result in statistically significant. If we have large enough sample sizes we can find statistically significant differences between groups which have no practical significance. This is like the twix bar from the left and right factories.

Practical and statistical significance are two different things entirely. With large data sets we can find statistically significant differences which may amount to NO practical significance.

As a matter of point if we have small sample sizes we can fail to find differences between groups as well. For example, if we had some nefarious wish to show that Turtles sizes on golf courses were no different than normal turtles, we might collect the data for only a few turtles (say 10) and then do a two tailed hypothesis test. Here is a simulation of this scenario:

fake.golf.course.turtles <- rnorm(10, mean=20.0, sd=8.0)
t.test(fake.golf.course.turtles, mu=19.0, alternative = "two.sided" )
##
##  One Sample t-test
##
## data:  fake.golf.course.turtles
## t = -1.35, df = 9, p-value = 0.21
## alternative hypothesis: true mean is not equal to 19
## 95 percent confidence interval:
##   9.557257 21.384448
## sample estimates:
## mean of x
##  15.47085

We can see that the mean of the golf course turtles mass is actually different than the population mean ($$\mu=19.0$$), however since our sample size is so small we will in general get large p values. Therefore, we fail to reject the null hypothesis. This is why we don’t say that we “accept the null hypothesis”, because failing to reject the null hypothesis might occur because there truly is no difference OR it may occur because we didn’t collect enough data. We can never prove the null hypothesis as being true, only collect and measure (using a p value) the evidence against it.

The golf course owner might misinterpret the results of the statistical test and report that the results of the turtle survey concluded that no difference existed between the golf course turtles and normal turtles.

We are careful to say we retain the null hypothesis or if you are a fan of double negative we fail to reject the null hypothesis because when we retain the null hypothesis this could happen for two reasons:

• The null hypothesis is in fact true
• We have insufficient data to reject the null hypothesis
Generally, we have no way of knowing which of these two reasons are the culprit.
Beware of the fallacy of accepting the null hypothesis. Under this fallacy we could prove anything by just never collecting enough evidence to reject the null hypothesis.

## 14.7 Hypothesis Testing for Population Fraction

We have seen how to perform hypothesis testing for the population mean $$\mu$$. We can also easily do hypothesis testing for the population proportion in R. The same principles apply to this case, except now we are testing for evidence that our samples proportion $$p$$ differs from a larger population.

### 14.7.1 Example:

Suppose we flip a coin 20 times and obtain 15 heads and 5 tails. Can we conclude that the coin is biased? Here are alternative hypothesis is that $$H_a: p \neq 0.5$$ and therefore the null hypothesis is $$H_0=0.5$$. To run this test in R we use the prop.test command:

prop.test(15,20, p=0.50, alternative="two.sided")
##
##  1-sample proportions test with continuity correction
##
## data:  15 out of 20, null probability 0.5
## X-squared = 4.05, df = 1, p-value = 0.04417
## alternative hypothesis: true p is not equal to 0.5
## 95 percent confidence interval:
##  0.5058845 0.9040674
## sample estimates:
##    p
## 0.75

You will notice that the syntax for this command is similar to the t.test hypothesis test. Here the first entry is the number of successes in our sample, the second entry is the total number of trials in the sample, p sets the null hypothesis value (like mu in t.test), and the alternative option is the same as the t.test.

The conditions for performing a hypothesis test for a population proportion are the same as forming a confidence interval. We need….

• At least five sucesses in our data set
• At least five failures in our data set
I call this the rule of 5 and 5.
Exercise 14.4 If I flip a coin 35 times and obtain 20 heads. If $$p$$ is the fraction of heads produced when flipping the coin can I conclude at a $$\alpha=0.05$$ significance level that the coin is biased towards heads?
Exercise 14.5 If only 5% of people have a rare genetic disease in the United States, if we find that 15 out of 100 people randomly sampled in Kerrville have the disease can we conclude with a significance level of $$\alpha=0.01$$ that Kerrville has a higher rate of this disease than the national average?

## 14.8 Hypothesis Testing in Linear Regression

We have seen some t tests being performed already in this class when we learned about linear regression. Let’s consider the simple example using the alligator data set. Recall this data set has two columns lnLength and lnWeight which gives the snout length of alligators against their weight. When plotting these two variable together we get a nice linear relationship in the scatter plot.

data("alligator")
plot(alligator$lnLength, alligator$lnWeight, xlab='Snout Length', ylab='Weight', main="Alligators")

Therefore, we are led to doing a linear regression analysis of the relationship between these variables. From this analysis we get the familiar summary.

lm.alligator=lm(lnWeight~lnLength, data=alligator)
summary(lm.alligator)
##
## Call:
## lm(formula = lnWeight ~ lnLength, data = alligator)
##
## Residuals:
##      Min       1Q   Median       3Q      Max
## -0.24348 -0.03186  0.03740  0.07727  0.12669
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  -8.4761     0.5007  -16.93 3.08e-10 ***
## lnLength      3.4311     0.1330   25.80 1.49e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.1229 on 13 degrees of freedom
## Multiple R-squared:  0.9808, Adjusted R-squared:  0.9794
## F-statistic: 665.8 on 1 and 13 DF,  p-value: 1.495e-12

However, now that we know a bit about hypothesis testing take a look at the last two columns in the Coefficients section. One column has t values and the next column gives us p values (3.08e-10, 1.49e-12). These p values are the result of performing a two tailed t test on the coefficients. These tell us if we have sufficient evidence to reject the null hypothesis that the coefficient is equal to zero.

The other p value that you can see in this summary is on the very last line. This p value is produced by a new type of hypothesis test (rather inventively called an F-test). An F-test for a regression model allows us to compare the fit of our model against a model with all the slopes set to zero. The null hypothesis of the F-test is that a model with no slope $$y=\alpha$$ can explain our data just as well as our linear model $$y=\beta x + \alpha$$.

In this simple case notice the F-test is redundant with the t test for the slope. In fact they give the exact same p value. In multiple regression models the F-test won’t be redundant with one of the slope t tests and gives us a way to assess the overall fit of our model.

We can also use the F-test to compare two nested regression models. By nested we mean where one of the models can be generated from the other by setting some of the slopes to zero. We won’t go into this in detail here. However if interested you can read more here.

## 14.9 Power of a Statistical Test

The final note for this section is to define the power of a statistical hypothesis test. The power of a test range from 0 to 1, with 1 being the most powerful test. The statistical power of a hypothesis test is given by $\text{Power}=1-\beta$ where $$\beta$$ is the Type II (False Negative) error rate. The power of a test a measurement of the ability of the test to distinguish truth from illusions. We saw that if we collect only a small amount of data, then we have very little evidence from which to draw our conclusions. Therefore, our statistical test has little power in this case. In addition, since collecting more data will in general decrease our Type II error rate we can increase the power of any test by collecting more data.

In Statistics much effort is devoted to finding the most powerful test available for a given scenario. Generally, to increase the power of a test without collecting more data we will need to add some assumptions about the data. If these are justified then all is well and we will get reduced error rates.

## 14.10 Homework

### 14.10.1 Concept Questions:

1. If I test 100 hypotheses using a significance level of $$\alpha=0.05$$, what is the expected number of Type I (False Positive) errors?
2. Collecting larger samples lowers both the Type I and Type II error rates for a Hypothesis test (True or False?) Why?
3. By testing a hypothesis many times against data, eventually I will collect a sample which allows me to reject the null hypothesis (True/False)
4. The p value measures the weight of evidence against the null hypothesis present in the data. (True/False)
5. The p value of a test is a random variable. (True/False)
6. By collecting enough data we can prove the null hypothesis (True/False)
7. Decreasing the significance parameter $$\alpha$$ will decrease the risk of False Negatives (Type II errors) (True/False)
8. A p value is the probability that the null hypothesis is true. (True/False)
9. A p value can be thought of as a conditional probability $$P(D|H_0)$$ where D is the data. (True/False)

### 14.10.2 Practice Problems:

For each of the scenarios below form the null and alternative hypothesis.

1. We want to measure if people in Kerrville consume on average more fried food in a week than the national average of 1 pound of fried food a week.
2. We would like to evaluate if a smaller fraction of people in Kerrville support banning those on the no-fly list from purchasing a gun than the national average. A national poll found that around 80% of the public supported this measure.
3. You have had have collected the number of cyber attacks each week on your website during the last year. You would like to test whether there has been a change in the long-time average of $$425$$ attacks per year.

td<-na.omit(Distel_Turtle_Data$Mass[Distel_Turtle_Data$Property=='BGAD'])
4. Load the wine_tasting data set. Conduct a hypothesis test to determine if the mean price of a bottle of wine from France is truly greater than the average price of a bottle of wine from all countries in the data set (the mean of the price column). Report and interpret the p value obtained from your test.
5. Load the NBA_Salaries_2017 data set. Determine whether we have sufficient evidence to conclude that the average guaranteed salary of NBA players ( NBA_Salaries_2017\$Guaranteed) exceeded 16 million in 2017. Report and interpret the p value obtained as well as justifying the use of the statistical hypothesis test chosen.


text:
## University Calculus: Early Transcendentals (3rd Edition)

Since the given series is a geometric series with common ratio $r=\dfrac{1}{10}$ Thus, for this series $|r|=\dfrac{1}{10} \lt 1$ Hence, the given series converges.


text:
Salta al contenuto principale

# ALESSANDRO BUSACCA

## PPG Embedded System for Blood Pressure Monitoring

• Autori: Agrò, D; Canicattì, R; Tomasino, A; Giordano, A; Adamo, G; Parisi, A; Pernice, R; Stivala, S; Giaconia, C; Busacca, A; Ferla, G
• Anno di pubblicazione: 2014
• Tipologia: Contributo in atti di convegno pubblicato in volume

### Abstract

In this work, we have designed and implemented a microcontroller-based embedded system for blood pressure monitoring through a PhotoPlethysmoGraphic (PPG) technique. In our system, it is possible to perform PPG measurements via reflectance mode. Hardware novelty of our system consists in the adoption of Silicon PhotoMultiplier detectors. The signal received from the photodetector is used to calculate the instantaneous heart rate and therefore the heart rate variability. The obtained results show that, by using our system, it is possible to easily extract both the PPG and the breath signal. These signals can be used to monitor the patients during the convalescence both in hospital and at home