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# Implications of mean-variance relationships for standardized mean differences

I spend more time than I probably should discussing meta-analysis problems on the R-SIG-meta-analysis listserv. The questions that folks pose there are often quite interesting—especially when they’re motivated by issues that they’re wrestling with while trying to complete meta-analysis projects in their diverse fields. For those interested in meta-analytic methodology, I think perusing the mailing list is a good way to get a bit of ground sense about problems that come up in practice and places where there is a need for new methodological work, or at least further methodological guidance.

Recently, a question came up on the listserv about whether it was reasonable to use the standardized mean difference metric for synthesizing studies where the outcomes are measured as proportions. Luke Martinez wrote:

I’m doing a meta-analysis where the papers report only “mean” and “sd” of some form of proportion and/or “mean” and “sd” of corresponding raw frequencies. (For context, the papers ask students to read, find, and correct the wrong words in a text.) … My question is given that all these studies only report “mean” and “sd”, can I simply use a SMD effect size?

I think this is an interesting question because, while the SMD could work perfectly fine as an effect size metric for proportions, there are also other alternatives that could be considered, such as odds ratios or response ratios or raw differences in proportions. Further, there are some situations where the SMD has disadvantages for synthesizing contrasts between proportions. Thus, it’s a situation where one has to make a choice about the effect size metric, and where the most common metric (the SMD) might not be the right answer. As I wrote in reply:

I would suggest that you could also consider other effect measures besides the SMD. For example, the response ratio is also a scale-free metric that could work with the proportion outcomes that you’ve described, and would also be appropriate for raw frequency counts as long as the total number possible is the same for the groups being compared within a given study.

Whether the response ratio would be more appropriate than the SMD is hard to gauge. One would need to know more about how the proportions were assessed and how the assessment procedures varied from study to study. For instance, did some studies use passages with many possible errors to be corrected while other studies used passages with just a few errors? Did the difficulty of the passages differ from study to study? Were there very low or very high mean proportions in any studies? Does there seem to be a relationship between the means and the variances of the proportions of a given group?

In a follow-up, I elaborated on some potential problems with using the SMD:

• Variation in the number of possible errors (and perhaps also in the length of the time provided for the test?) suggests that the measures from different studies may have varying degrees of reliability. Varying reliability introduces heterogeneity in the SMD (because the denominator is inflated or shrunk by the degree of reliability).

• A relationship between the M and SD of the proportions for a given group suggests that the distribution of the individual-level outcomes might also exhibit mean-variance relationships. (I say “suggests” rather than implies because there’s an ecological inference here, i.e., assuming something about individual-level variation on the basis of group-level variation.) If this supposition is reasonable, then that introduces a further potential source of heterogeneity in the SMDs (study-to-study variation in the M for the reference group influences the SD of the reference group, thereby inflating or shrinking the SMDs).

And I suggested a possible work-flow for examining the choice of effect size metric:

Here’s how I might proceed if I were conducting this analysis:

1. Calculate both SMDs and log-transformed response ratios for the full set of studies.
2. Examine the distribution of effect size estimates for each metric (using histograms or funnel plots). If one of the distributions is skewed or has extreme outliers, take that as an indication that the metric might not be appropriate.
3. Fit meta-analytic models to summarize the distribution of effect sizes in each metric, using a model that appropriately describes the dependence structure of the estimates. Calculate I-squared statistics, give preference to the metric with lower I-squared.
4. If (2) and (3) don’t lead to a clearly preferable metric, then choose between SMD and RR based on whichever will make the synthesis results easier to explain to people.
5. (Optional/extra credit) Whichever metric you choose, repeat your main analyses using the other metric and stuff all those results in supplementary materials, to satisfy any inveterate statistical curmudgeons who might review/read your synthesis.

(When I referred to “inveterate statistical curmudgeons”, I mostly had myself in mind.)

In this post, I want to provide a bit more detail regarding why I think mean-variance relationships in raw data can signal that the standardized mean differences might be less useful as an effect size metric compared to alternatives. The concern is actually broader than meta-analyses of outcomes measured as proportions, so I’ll start with a different case and then return to a situation similar to the one described in the original question.

## Mean-variance relationships can induce heterogeneity

The standardized mean difference parameter for a given study can be defined as: $\delta_i = \frac{\mu_{Bi} - \mu_{Ai}}{\sigma_{Ai}},$ where $$\mu_{Ai}$$ and $$\mu_{Bi}$$ are the (population) mean outcomes in group $$A$$ and group $$B$$ of study $$i$$ and $$\sigma_{Ai}$$ is the (population) standard deviation in group $$A$$ of study $$i$$.1 The ideal case for using the SMD metric is when the outcomes in different studies are linearly equatable, so that the outcome scale in one study can be directly translated into the outcome scale of another study. However, if outcomes exhibit mean-variance relationships, linearly equatability seems rather implausible, and we might expect that SMDs will display heterogeneity across studies as a result.

Let me lay out an example of a situation where the outcomes exhibit mean-variance relationships and where, as a consequence, the SMD metric becomes heterogeneous. Suppose that we have $$k$$ studies, each involving a two-group comparison, with groups of equal size. In study $$i$$, the outcomes in group $$A$$ follow a poisson distribution with mean $$\mu_{Ai}$$, so that the variance of the outcomes in group $$A$$ is also $$\mu_{Ai}$$, for $$i = 1,...,k$$. The outcomes in group $$B$$ follow a poisson distribution with mean $$\mu_{Bi}$$, so the variance is also $$\mu_{Bi}$$. Now, suppose that there is a fixed, proportional relationship between $$\mu_{Bi}$$ and $$\mu_{Ai}$$, so that $$\mu_{Bi} = \lambda \mu_{Ai}$$ for some $$\lambda > 0$$. In other words, the treatment contrast is constant on the scale of the response ratio. However, the means in group $$A$$ vary from study to study. To make things concrete, let’s assume that the means in group $$A$$ follow a gamma distribution with shape parameter $$\alpha$$ and rate parameter $$\beta$$: $\mu_{Ai} \sim \Gamma(\alpha, \beta).$ What does this model imply about the distribution of standardized mean differences across this set of studies?

Under this model, the SMD parameter for study $$i$$ is: $\delta_i = \frac{\mu_{Bi} - \mu_{Ai}}{\sqrt{\mu_{Ai}}} = (\lambda - 1) \times \sqrt{\mu_{Ai}}.$ The first term in the above expression is a constant that only depend on the size of the response ratio, but the second term is random because we have assumed that the group $$A$$ means vary from study to study. It will therefore create heterogeneity in the SMD parameters—the greater the variance of the $$\mu_{Ai}$$’s, the greater the heterogeneity in $$\delta_i$$. Specifically, under the above assumptions, the effect size parameters follow a Nakagami distribution: $\delta_i \sim \text{Nakagami}\left(m = \alpha, \Omega = \frac{(\lambda - 1)^2 \alpha}{\beta}\right)$ Thus, even though we have a model where there is an underlying fixed relationship between $$\mu_{Ai}$$ and $$\mu_{Bi}$$, using the SMD metric for synthesis will lead to a situation with heterogeneous effects (even if all of the studies had large sample sizes and so effect sizes in individual studies are precisely estimated).

## An example with proportions

This sort of behavior is not restricted to the poisson-gamma model I sketched above. The key features of that example are a) the assumption that the outcomes have a strong mean-variance relationship and b) the assumption that the $$\mu_{Ai}$$’s are heterogeneous across studies. If both of these hold, then the resulting SMDs will also be heterogeneous. I’ll now describe a similar model, but where the outcomes within each study are proportions.

As before, suppose that we have $$k$$ studies, each involving a two-group comparison, with groups of equal size. In study $$i$$, the outcomes in group $$A$$ follow a binomial distribution with mean proportion $$\pi_{Ai}$$ and $$T_i$$ trials, so that the variance of the outcomes in group $$A$$ is $$\pi_{Ai}\left(1 - \pi_{Ai}\right) T_i$$, for $$i = 1,...,k$$. The outcomes in group $$B$$ also follow a binomial distribution, this one with mean proportion $$\pi_{Bi}$$ and $$T_i$$ trials, so the variance is $$\pi_{Bi}\left(1 - \pi_{Bi}\right) T_i$$. Next, to induce variation in the group-$$A$$ means, let’s assume that the mean proportions follow a beta distribution: $\pi_{Ai} \sim \text{Beta}(\alpha, \beta).$

Finally, suppose that $$\pi_{Bi} = \lambda_i \pi_{Ai}$$ for some $$\lambda_i > 0$$.

Under these assumptions, the SMD parameter for study $$i$$ is: $\delta_i = \frac{\pi_{Bi}T_i - \pi_{Ai} T_i}{\sqrt{\pi_{Ai} (1 - \pi_{Ai}) T_i}} = (\lambda_i - 1) \times \sqrt{T_i} \times \sqrt{\frac{\pi_{Ai}}{1 - \pi_{Ai}}}.$ From the above expression, it can be seen that there are three potential sources of variation in $$\delta_i$$: variation in the study-specific response ratio $$\lambda_i$$, variation in the group-$$A$$ proportions $$\pi_{Ai}$$, and variation in the number of trials $$T_i$$. The total heterogeneity in $$\delta_i$$ will depend on all three, as well as on the co-variation between $$\lambda_i$$, $$\pi_{Ai}$$, and $$T_i$$.

To make this concrete, let me simulate some meta-analytic data that follows the above model. To do so, I’ll need to make some additional distributional assumptions2:

1. that $$\lambda_i$$ is log-normally distributed such that $$\ln \lambda_i \sim N(\ln \Lambda, \tau^2)$$;
2. that the number of trials is uniformly distributed on the integers between $$t_{min}$$ and $$t_{max}$$;
3. that $$N_i$$, the number of observations per group in study $$i$$, is uniformly distributed on the integers between $$n_{min}$$ and $$n_{max}$$; and
4. that $$\pi_{Ai}$$, $$\lambda_i$$, $$T_i$$, and $$N_i$$ are mutually independent.

Here’s a function that generates study-specific parameter values and sample proportions:

sim_binom_summary <- function(pi_i, T_i, n_i) {
y <- rbinom(n_i, size = T_i, prob = pi_i) / T_i
data.frame(M = mean(y), SD = sd(y))
}

sim_props <- function(
k, # number of studies
alpha, beta, # parameters of pi_Ai distribution,
Lambda, tau, # parameters of lambda_i distribution
t_min, t_max, # parameters of T_i distribution
n_min, n_max # parameters of the sample size distribution
) {

# simulate parameters
pi_Ai <- rbeta(k, shape1 = alpha, shape2 = beta)
lambda_i <- exp(rnorm(k, mean = log(Lambda), sd = tau))
pi_Bi <- lambda_i * pi_Ai
T_i <- sample(t_min:t_max, size = k, replace = TRUE)
delta_i <- (pi_Bi - pi_Ai) * T_i / sqrt(pi_Ai * (1 - pi_Ai) * T_i)
n_i <- sample(n_min:n_max, size = k, replace = TRUE)

# simulate data
stats_A <- purrr::pmap_dfr(list(pi_i = pi_Ai, T_i = T_i, n_i = n_i),
sim_binom_summary)

stats_B <- purrr::pmap_dfr(list(pi_i = pi_Bi, T_i = T_i, n_i = n_i),
sim_binom_summary)

# compile
res <- data.frame(
pi_Ai = pi_Ai, pi_Bi = pi_Bi,
lambda_i = lambda_i, T_i = T_i,
delta_i = delta_i, n_i = n_i,
mA = stats_A$M, sdA = stats_A$SD,
mB = stats_B$M, sdB = stats_B$SD
)

# effect size calculations
res <- metafor::escalc(
data = res, measure = "ROM", var.names = c("lRR", "V_lRR"),
m1i = mB, m2i = mA,
sd1i = sdB, sd2i = sdA,
n1i = n_i, n2i = n_i
)
res <- metafor::escalc(
data = res, measure = "SMD", var.names = c("d", "V_d"),
m1i = mB, m2i = mA,
sd1i = sdB, sd2i = sdA,
n1i = n_i, n2i = n_i
)

res
}

set.seed(20211024)
dat <- sim_props(k = 60, alpha = 12, beta = 4,
Lambda = 0.7, tau = .05,
t_min = 5, t_max = 18,
n_min = 10, n_max = 40)

head(dat)
##       pi_Ai     pi_Bi  lambda_i T_i    delta_i n_i        mA       sdA
## 1 0.7584480 0.5836965 0.7695933  11 -1.3540950  24 0.7500000 0.1080650
## 2 0.7359047 0.4950740 0.6727420  16 -2.1851474  24 0.7786458 0.1222235
## 3 0.7132014 0.4773027 0.6692398  12 -1.8068471  10 0.7333333 0.1097134
## 4 0.6223653 0.4627406 0.7435193   9 -0.9877857  30 0.6666667 0.1399386
## 5 0.5916619 0.4205407 0.7107787   6 -0.8527716  28 0.5833333 0.2103299
## 6 0.7266748 0.5014601 0.6900751   9 -1.5160305  35 0.7619048 0.1209466
##          mB       sdB     lRR  V_lRR       d    V_d
## 1 0.6174242 0.1285066 -0.1945 0.0027 -1.0983 0.0959
## 2 0.5260417 0.1275776 -0.3922 0.0035 -1.9888 0.1245
## 3 0.3583333 0.1622089 -0.7161 0.0227 -2.5934 0.3681
## 4 0.4555556 0.1943213 -0.3808 0.0075 -1.2306 0.0793
## 5 0.4583333 0.2060055 -0.2412 0.0119 -0.5921 0.0746
## 6 0.4920635 0.1793349 -0.4372 0.0045 -1.7447 0.0789

For the specified parameter values, there is only a small amount of true heterogeneity in the log of the response ratios (the blue density). Of course, there is further heterogeneity in the log response ratio estimates (the green density) due to sampling error:

library(ggplot2)
ggplot(dat) +
geom_density(aes(log(lambda_i), ..scaled..), fill = "blue", alpha = 0.5) +
geom_density(aes(lRR, ..scaled..), fill = "green", alpha = 0.2) +
theme_minimal()

A random effects meta-analysis confirms that there is only a modest degree of true heterogeneity in the log response ratios:

library(metafor)
rma(yi = lRR, vi = V_lRR, data = dat)
##
## Random-Effects Model (k = 60; tau^2 estimator: REML)
##
## tau^2 (estimated amount of total heterogeneity): 0.0028 (SE = 0.0013)
## tau (square root of estimated tau^2 value):      0.0529
## I^2 (total heterogeneity / total variability):   42.01%
## H^2 (total variability / sampling variability):  1.72
##
## Test for Heterogeneity:
## Q(df = 59) = 100.6304, p-val = 0.0006
##
## Model Results:
##
## estimate      se      zval    pval    ci.lb    ci.ub
##  -0.3498  0.0111  -31.5751  <.0001  -0.3715  -0.3281  ***
##
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Contrast this with what we get from using the standardized mean difference metric. The distributions of true effect sizes (blue) and of effect size estimates (light purple) have large spread as well as strong left skew:

library(ggplot2)
ggplot(dat) +
geom_density(aes(delta_i, ..scaled..), fill = "blue", alpha = 0.2) +
geom_density(aes(d, ..scaled..), fill = "purple", alpha = 0.5) +
theme_minimal()

A random effects meta-analysis of the standardized mean differences shows a greater degree of true heterogeneity, both in terms of the estimated $$\tau$$ and in $$I^2$$, or the proportion of total variance in the effect size estimates that is attributable to true heterogeneity:

rma(yi = d, vi = V_d, data = dat)
##
## Random-Effects Model (k = 60; tau^2 estimator: REML)
##
## tau^2 (estimated amount of total heterogeneity): 0.2838 (SE = 0.0743)
## tau (square root of estimated tau^2 value):      0.5327
## I^2 (total heterogeneity / total variability):   72.61%
## H^2 (total variability / sampling variability):  3.65
##
## Test for Heterogeneity:
## Q(df = 59) = 203.0513, p-val < .0001
##
## Model Results:
##
## estimate      se      zval    pval    ci.lb    ci.ub
##  -1.5967  0.0824  -19.3771  <.0001  -1.7582  -1.4352  ***
##
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

## Diagnostics

The code above more-or-less implements the workflow I suggested for deciding between the standardized mean difference or response ratio metric (for proportions, we could also add further comparisons with log odds ratios and with raw differences in proportions). But is there further diagnostic information in the data that could provide a better sense of what is going on? I think there are a few things that might be helpful to consider.

First, the issues I’m concerned with here will arise when there are mean-variance relationships in the outcomes. To get at that, we can simply plot the means and SDs of each group. In the code below, I re-structure the data so that there is one row per group per study. I then plot the SD versus the mean of each group:

library(dplyr)
library(tidyr)

long_summary_stats <-
dat %>%
select(n_i, T_i, mA, sdA, mB, sdB) %>%
pivot_longer(cols = c(mA, sdA, mB, sdB),
names_to = c(".value","group"),
names_pattern = "(m|sd)(A|B)")

ggplot(long_summary_stats,
aes(m, sd, color = group)) +
geom_point() +
geom_smooth(se = FALSE) +
scale_x_continuous(limits = c(0, 1), expand = c(0,0)) +
expand_limits(y = 0) +
theme_minimal() +
theme(legend.position = c(0.1, 0.9))

The plot above does suggest a mean-variance relationship, though it’s a bit messy. We can do better by using the scaled SD, after adjusting for the degree of spread that we would expect given $$T_i$$:

long_summary_stats %>%
mutate(
sd_scaled = sd * sqrt(T_i)
) %>%
ggplot(aes(m, sd_scaled, color = group)) +
geom_point() +
geom_smooth(se = FALSE) +
geom_function(fun = function(x) sqrt(x * (1 - x)),
color = "black") +
scale_x_continuous(limits = c(0, 1), expand = c(0,0)) +
expand_limits(y = 0) +
theme_minimal() +
theme(legend.position = c(0.1, 0.9))

From the above, it does appear that there could be a relationship between the scaled SD and the mean. The black curve indicates the theoretical mean-variance relationship that would be expected under the binomial distribution, and indeed the empirical relationship appears to be quite similar. This suggests that mean-variance relationships might be at play (a correct supposition, since of course we know the true data-generating process here).

Second, since the outcomes in each group are all proportions, we can simply plot the mean in group $$B$$ versus the mean in group $$A$$:

ggplot(dat, aes(mA, mB)) +
geom_point() +
geom_smooth(se = FALSE, color = "green") +
geom_smooth(method = "lm", formula = y ~ x) +
coord_cartesian(xlim = c(0,1), ylim = c(0,1), expand = FALSE) +
theme_minimal()

This plot shows that there is a strong linear relationship between the two means, with a best-fit line that might go through the origin. This suggests that the response ratio might be an appropriate metric (although the difference in proportions might also be appropriate here, since a line with unit slope would probably fit quite well).

Third (and most speculatively/hand-wavily), I think exploratory moderator analysis can be useful here, but interpreted in a non-typical way. Under the model I’ve sketched, we would expect that the standardized mean difference estimates should be systematically associated with the group-$$A$$ means, as well as with the number of trials used to assess outcomes. The scatter-plots below show that this is indeed the case (the right-hand plot shows $$d_i$$ versus $$\sqrt{T_i}$$).

library(patchwork)
mA_d_plot <-
ggplot(dat, aes(mA, d)) +
geom_point() +
geom_smooth(se = FALSE, color = "green") +
geom_smooth(method = "lm") +
scale_x_continuous(limits = c(0, 1), expand = c(0,0)) +
theme_minimal()

Ti_d_plot <-
ggplot(dat, aes(sqrt(T_i), d)) +
geom_point() +
geom_smooth(se = FALSE, color = "green") +
geom_smooth(method = "lm") +
theme_minimal()

mA_d_plot + Ti_d_plot

This impression is also born out by a meta-regression that includes the group-$$A$$ means and $$\sqrt{T_i}$$ as moderators:

rma(d ~ mA + sqrt(T_i), vi = V_d, data = dat)
##
## Mixed-Effects Model (k = 60; tau^2 estimator: REML)
##
## tau^2 (estimated amount of residual heterogeneity):     0.0238 (SE = 0.0238)
## tau (square root of estimated tau^2 value):             0.1544
## I^2 (residual heterogeneity / unaccounted variability): 18.12%
## H^2 (unaccounted variability / sampling variability):   1.22
## R^2 (amount of heterogeneity accounted for):            91.60%
##
## Test for Residual Heterogeneity:
## QE(df = 57) = 68.9706, p-val = 0.1330
##
## Test of Moderators (coefficients 2:3):
## QM(df = 2) = 110.9125, p-val < .0001
##
## Model Results:
##
##            estimate      se     zval    pval    ci.lb    ci.ub
## intrcpt      2.5225  0.3964   6.3632  <.0001   1.7455   3.2995  ***
## mA          -2.8326  0.4336  -6.5321  <.0001  -3.6825  -1.9827  ***
## sqrt(T_i)   -0.6109  0.0756  -8.0855  <.0001  -0.7590  -0.4628  ***
##
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Here are the same plots as above, but using the log of the response ratio as the effect size metric:

mA_lRR_plot <-
ggplot(dat, aes(mA, lRR)) +
geom_point() +
geom_smooth(se = FALSE, color = "green") +
geom_smooth(method = "lm") +
scale_x_continuous(limits = c(0, 1), expand = c(0,0)) +
theme_minimal()

Ti_lRR_plot <-
ggplot(dat, aes(sqrt(T_i), lRR)) +
geom_point() +
geom_smooth(se = FALSE, color = "green") +
geom_smooth(method = "lm") +
theme_minimal()

mA_lRR_plot + Ti_lRR_plot

In the left-hand plot, there does not appear to be any relationship between the effect size estimates and the group-$$A$$ means. In the right-hand plot, there does seem to be a mild relationship between the effect size estimates and $$\sqrt{T_i}$$, which is a bit surprising, although the strength of the relationship is much weaker than what we saw with the standardized mean differences. Meta-regression analysis supports these interpretations:

rma(lRR ~  mA + sqrt(T_i), vi = V_lRR, data = dat)
##
## Mixed-Effects Model (k = 60; tau^2 estimator: REML)
##
## tau^2 (estimated amount of residual heterogeneity):     0.0019 (SE = 0.0011)
## tau (square root of estimated tau^2 value):             0.0439
## I^2 (residual heterogeneity / unaccounted variability): 32.87%
## H^2 (unaccounted variability / sampling variability):   1.49
## R^2 (amount of heterogeneity accounted for):            31.30%
##
## Test for Residual Heterogeneity:
## QE(df = 57) = 84.4977, p-val = 0.0105
##
## Test of Moderators (coefficients 2:3):
## QM(df = 2) = 10.6344, p-val = 0.0049
##
## Model Results:
##
##            estimate      se     zval    pval    ci.lb    ci.ub
## intrcpt     -0.2362  0.0950  -2.4864  0.0129  -0.4224  -0.0500   *
## mA           0.1061  0.0948   1.1196  0.2629  -0.0796   0.2918
## sqrt(T_i)   -0.0553  0.0179  -3.0852  0.0020  -0.0904  -0.0202  **
##
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Now, you might think that a meta-analyst should get excited about the standardized mean difference results, since they’ve uncovered two systematic predictors of effect size magnitude. However, both of these factors are purely operational, arbitrary features of the (simulated) study designs, rather than theoretically or substantively interesting features of the studies. Considered in this light, the finding that they each moderate the magnitude of the standardized mean differences is, more than anything else, annoying. If we wanted to examine other more theoretically interesting moderators, we’d have to do so in a way that accounts for these methodological predictors. At minimum, that would mean including them all in a meta-regression (leading to a model with 3+ predictors). Further, we would have to worry about whether the functional form of the regression is reasonable. Simply adding the theoretical moderator to the model amounts to assuming that it predicts effect size magnitude in a linear, additive fashion, but what if that’s not the right model? Since we know the true data-generating process here, we can see that the linear, additive model would not be correct. But in practice, when we don’t know the true process, this would be much murkier.

The general principle that I’m suggesting here is that effect sizes should ideally be on a metric that is independent of arbitrary methodological factors because this should reduce overall heterogeneity and simplify the model, making it easier to detect real relations of interest. If one has a choice between several different effect size metrics, then a metric that shows clear associations with methodological factors should be discounted in favor of metrics that do not show such associations or show them only weakly. How to fully operationalize this sort of decision (as one would need to when writing a protocol for a meta-analysis, for example), I’m not yet sure about. It seems like a useful avenue for further methodological work.

1. Yes, there are other ways to define the SMD. Yes, usually we use the standard deviation pooled across both groups. I’m going to use the standard deviation in group $$A$$ alone because it simplifies some of the mathy bits. Please feel free to work through the case with a pooled SD for yourself.↩

2. One of the vexing things about simulations is that you often end up needing to specify a bunch of assumptions about auxiliary quantities, beyond those of the model you’re actually interested in investigating.↩


text:
# Module 1: Kinematics | Beginner’s Guide to Year 11 Physics

Get your marks into motion with this article explaining Year 11 Physics Module 1, Kinematics. We discuss the 3 topics and explain how to solve key problems.

Struggling to get the ball rolling with your Physics marks? In this Guide, we’ll walk you through Module 1 Kinematics for year 11 Physics so your marks get the right kind of momentum!

## What is kinematics?

Before we study the physics behind how objects behave and interact, we need to establish a precise system of language and mathematics to describe our observations. In Year 11, we will be using more technical language compared to previous years; some words which have a broad meaning in everyday life (such as ‘energy’ and ‘work’) have a specific meaning in the context of physics, and so we must ensure we all agree on their definitions.

Kinematics is the description and analysis of the motion of objects without studying the cause of the motion.

We will start by introducing the concept of vector quantities. From there, we will apply our knowledge to describing one-dimensional motion, followed by expanding to motion on a plane.

## Topic 1: Vectors

In physics, we deal with a variety of quantities such as mass, distance, energy, etc.

Physical quantities can be classified into two main categories:

• Scalars: A scalar is any quantity that has a magnitude (amount or size) but does not have a direction. A scalar requires only a number and the units to be complete. Mass and area are examples of scalar quantities, e.g. Australia has an area of 7,617,930 km2.
• Vectors: A vector is any quantity that has a magnitude and a direction associated with it. We say a city is 1000 km south or a car travels at 60 kmh-1 east.

In kinematics, the vectors of interest are:

• Displacement $$\vec s$$: the change in position of an object, or the difference between where it started and where it ended.
• Velocity $$\vec v$$: how quickly or slowly an object’s displacement is changing
$$\vec v=\frac{Δ\vec s}{Δt}$$
• Acceleration $$\vec a$$: the rate of change of an object’s velocity
$$\vec a= \frac{∆\vec v}{∆t}=\frac{\vec v-\vec u}{∆t}$$
where $$v$$ is the final velocity and $$u$$ is the initial velocity.

All of these quantities have a magnitude and direction, and they are mathematically related to each other as we will see in this guide.

Displacement has a scalar counterpart, distance. Distance only measures the total amount of ‘ground’ covered, with no regard for direction. It is possible for an object’s distance travelled to be very different to the magnitude of its final displacement.

Velocity’s scalar counterpart is speed. Acceleration does not have a specific counterpart.

In order to work with vector quantities, we must first describe how they can be interpreted and mathematically manipulated (e.g. addition and subtraction). We can represent vectors as arrows, with the length of the line representing the magnitude of the quantity and the head of the arrow pointing in the required direction.

When adding two vectors, such as two displacement vectors $$\vec s_1+ \vec s_2$$, the vectors are joined head to tail.

The tail of the second vector joins the head of the first vector.

The resultant vector or final vector spans from the tail of the first vector to the head of the second vector.

For example: (10 m right) + (20 m right) = (30 m right)

The vector subtraction $$\vec s_1- \vec s_2$$ is equivalent to the vector addition $$\vec s_1+ (-\vec s_2)$$.
The negative sign reverses the direction of $$\vec s_2$$.

For example: (10 m right) – (15 m left) is equivalent to (10 m right) + (-15 m left) = (10 m right) + (15 m right) = (25 m right)

Vectors can also be added and subtracted in two dimensions. We will discuss this in section 3.

A vector can be divided or multiplied by a scalar – for example, displacement is divided by time to get velocity – and the result is another vector in the same direction.

Vector operations beyond those described here are covered in the Year 12 Mathematics Extension courses, and if you continue studying science and maths at university you will frequently use vectors as tools.

## Topic 2: Motion in a straight line

If our problem is restricted to motion in a straight line (which can be any line: vertical, horizontal or otherwise, such as the surface of an inclined plane), vectors can only point in one of two directions (e.g. up or down if the line is vertical).

We can solve problems mathematically by designating one direction to be positive and the other to be negative. All vectors that point in the positive direction are then associated with positive numbers, and all vectors pointing in the negative direction correspond to negative numbers.

In particular, when we have a situation involving constant acceleration, the following equations allow us to link acceleration, velocity, displacement and time:

\begin{align*}
v & =u+at\\
s &=ut+\frac{1}{2} at^2 \\
v^2 &=u^2+2as\\
\end{align*}

In these equations, $$u$$ represents the object’s initial velocity and $$v$$ is its final velocity.

We can also solve problems relating to the relative motion between two objects, or the resultant motion of an object in a moving medium. Consider a situation where an aeroplane is flying on a windy day. Its speed relative to the ground depends on its airspeed but also how fast the air is travelling relative to the ground.

For an object A travelling through a medium B moving relative to the ground G:

$$\vec v_{AG}=\vec v_{AB}+ \vec v_{BG}$$

where $$v_{AG}$$ is the velocity of A relative to G
$$v_{AB}$$ is the velocity of A relative to B and
$$v_{BG}$$ is the velocity of B relative to G.

In other questions, you will know the velocity of both objects relative to the ground and need to calculate their velocity relative to each other. In that case, rearrange the previous equation to find that

$$\vec v_{AB}= \vec v_{AG}- \vec v_{BG}$$

The challenging part of these questions is usually not the vector calculation, which you will have lots of practice with. It’s working out how the information in the question translates into vectors.

Consider the following scenario: a car travelling West at 100 kmh-1 drives past a truck travelling west at 70 kmh-1.

Velocity of the car relative to the ground $$v_{CG}=100 \text{ kmh}^{-1}$$ West.
Velocity of the truck relative to the ground: $$v_{CG}=70 \text{ kmh}^{-1}$$ West.
Velocity of the car relative to the truck $$v_{CT}=v_{CG}-v_{TG}=30 \text{ kmh}^{-1}$$ West.

Across all the sciences, graphs are used to summarise and convey information. Physics is no exception. If we have data about the displacement, velocity or acceleration of an object over time, we can convert it to a graph.

What does this graph tell us? The person starts at some origin point, then travels $$8 \text{ m}$$ to the North. Once there, she stays in place for two seconds, before starting to walk back towards the origin. At $$t = 10 \text{ s}$$, she is back in her original position.

We can do more than plot and read graphs. There is additional information available through analysing their features. For example, we have established that velocity is the rate of change of displacement. Therefore, if we take the gradient of a displacement-time graph, it represents the quantity $$\frac{Δ\vec s}{Δt}$$ which is equal to the instantaneous velocity. Similarly, we can take the gradient of a velocity-time graph to find the instantaneous acceleration.

One more thing we can do with a velocity-time graph is work the other way to calculate total displacement. This is found by taking the area under the graph: any area above the x-axis adds to the total positive displacement, and any area under the x-axis adds to the negative displacement.

## Topic 3: Motion on a plane

More generally, we want to be able to solve problems in two dimensions. Fortunately, most of our techniques remain the same. We just need to extend our understanding of vector operations first.

Adding vectors still works the same way: when adding two vectors, the vectors are joined head to tail. The resultant vector spans from the tail of the first vector all the way to the end of the final vector.

If the problem forms a triangle, you may be able to solve for unknown lengths and angles using the cosine and/or sine rules. More generally, especially if you are adding three or more vectors, the best option is to decompose each vector into horizontal and vertical components using trigonometry:

After decomposing, you can add the horizontal components separately, and also the vertical components separately to get the horizontal and vertical components of the final vector. Adding these results in a right angle triangle which you can solve to find the final vector.

There are a few ways to express answers when working in 2D. The most common are compass bearings (e.g. N30E) or true bearings (e.g. 030 T) which you should be familiar with from junior Geography. A word of caution: if you have used the cosine and sine rules to solve for the properties of the unknown vector, the angles inside the triangle are most likely not the angles you need to express your answer.

Almost everything we were able to do in 1D can be expanded to include 2D motion. The process of calculating relative and resultant velocities is the same – the vector operations are simply done in two dimensions instead of one. The equations of motion with constant acceleration can’t be used directly, but in Year 12 you will learn how to modify them to solve further motion problems.

The extra dimension also introduces an extra step to representing motion with graphs. There are now two coordinates that can change with time, and so to fully describe motion we must keep track of both. There are two possibilities: either we keep track of the $$x$$ and $$y$$ components motion over time, or we describe the vectors by their magnitude $$r$$ and bearing $$θ$$.

Imagine a car travelling at constant speed around a roundabout:

We could represent its motion like this:

or like this:

Both sets of plots contain the same information.

Kinematics may not initially seem hugely exciting, but it forms the foundation of our ability to describe everything from atoms to the universe. Once you master this module, you will find it easier to pick up and understand new physics, starting with Module 2: Dynamics.

© Matrix Education and www.matrix.edu.au, 2021. Unauthorised use and/or duplication of this material without express and written permission from this site’s author and/or owner is strictly prohibited. Excerpts and links may be used, provided that full and clear credit is given to Matrix Education and www.matrix.edu.au with appropriate and specific direction to the original content.

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text:
# A uniform momentum zone-vortical fissure model of the turbulent boundary layer

### Abstract

• Recent studies reveal that at large friction Reynolds number $\unicode[STIX]{x1D6FF}^{+}$ the inertially dominated region of the turbulent boundary layer is composed of large-scale zones of nearly uniform momentum segregated by narrow fissures of concentrated vorticity. Experiments show that, when scaled by the boundary-layer thickness, the fissure thickness is $\mathit{O}(1/\sqrt{\unicode[STIX]{x1D6FF}^{+}})$, while the dimensional jump in streamwise velocity across each fissure scales in proportion to the friction velocity $u_{\unicode[STIX]{x1D70F}}$. A simple model that exploits these essential elements of the turbulent boundary-layer structure at large $\unicode[STIX]{x1D6FF}^{+}$ is developed. First, a master wall-normal profile of streamwise velocity is constructed by placing a discrete number of fissures across the boundary layer. The number of fissures and their wall-normal locations follow scalings informed by analysis of the mean momentum equation. The fissures are then randomly displaced in the wall-normal direction, exchanging momentum as they move, to create an instantaneous velocity profile. This process is repeated to generate ensembles of streamwise velocity profiles from which statistical moments are computed. The modelled statistical profiles are shown to agree remarkably well with those acquired from direct numerical simulations of turbulent channel flow at large $\unicode[STIX]{x1D6FF}^{+}$. In particular, the model robustly reproduces the empirically observed sub-Gaussian behaviour for the skewness and kurtosis profiles over a large range of input parameters.
• ### Authors

• Bautista, Juan Carlos Cuevas
• White, Christopher
• Chini, Gregory
• Klewicki, Joseph C

### Publication Date

• November 6, 2018

### Keywords

• boundary layer structure
• turbulence modelling
• turbulent boundary layers

• 609

• 633

• 858


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# make_co_wd¶

This test builds a carbon-oxygen white dwarf starting from a specified pre main-sequence mass. It does this by running through a series of inlists for different stages of evolution. By default, the first two steps (pre main-sequence and main-sequence through core helium burning) are skipped so that the test will run in a reasonable amount of time, and the test produces a 0.6 $${\rm M}_\odot$$ white dwarf from a roughly 3 $${\rm M}_\odot$$ progenitor.

Starting from the beginning to produce a white dwarf descended from a progenitor of different mass or metallicity can be accomplished by editing the values assigned to initial_mass or initial_z in inlist_common.

Larger values for initial_mass can be used to produce more massive carbon-oxygen white dwarfs, but note that this test produces only an approximate initial-to-final-mass relation (IFMR). The test is designed to truncate the AGB evolution at the beginning of the thermally-pulsing AGB (TP-AGB) stage, which can be difficult and/or resource-intensive to model.

The five steps in this procedure are:

## 1. inlist_zams (optional)¶

This step starts from the pre main-sequence to produce a ZAMS model with the initial mass and metallicity specified in inlist_common.

## 2. inlist_to_end_he_core_burn (optional)¶

This step evolves from ZAMS through the end of core helium burning.

## 3. inlist_co_core¶

This step evolves to near the end of shell helium burning, producing the final C/O core profile in the interior. This step terminates when a thermal pulse occurs in the helium shell.

## 4. inlist_remove_env¶

This step removes the AGB envelope using the star_relax_to_star_cut method in src/run_star_extras.f90. The location of the cut is specified using x_ctrl(1) to leave $$10^{-3}$$ $${\rm M}_\odot$$ of the hydrogen envelope. After the relaxation ends, residual burning will reduce the final hydrogen envelope mass to around $$10^{-4}$$ $${\rm M}_\odot$$ (for the default 0.6 $${\rm M}_\odot$$ white dwarf).

## 5. inlist_settle¶

This step turns on diffusion in the young proto-WD model to allow the model to settle into a stratified envelope structure with a pure hydrogen atmosphere. This final step ends when the white dwarf has cooled down to reach a luminosity of 1 $${\rm L}_\odot$$.


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# Computer Science Theory at Northeastern

## Seminar

Click here to subscribe to our mailing list.
Click here to subscribe to our Google Calendar.

## Theory Seminar

This semester the theory seminar will be every Thursday from 12:00-1:30pm. We have reserved Forsyth 150, but Room 655 in ISEC will sometimes be available. The Theory Seminar will occur every week. If we do not have an invited speaker, we will have short presentations from students or faculty. Write to Jonathan Ullman with suggestions for speakers!

## Fall Semester, 2018

 September 1312:00PM150 Forsyth Eylon Yogev, Weizmann Institute of Science Search Problems: A Cryptographic Perspective Abstract The class TFNP is the search analog of NP with the additional guarantee that any instance has a solution. TFNP has attracted extensive attention due to its natural syntactic subclasses that capture the computational complexity of important search problems from algorithmic game theory, combinatorial optimization and computational topology. Thus, one of the main research objectives in the context of TFNP is to search for efficient algorithms for its subclasses, and at the same time proving hardness results where efficient algorithms cannot exist. Currently, no problem in TFNP is known to be hard under assumptions such as NP hardness, the existence of one-way functions, or even public-key cryptography. The only known hardness results are based on less general assumptions such as the existence of collision-resistant hash functions, one-way permutations less established cryptographic primitives (e.g. program obfuscation or functional encryption). Several works explained this status by showing various barriers to proving hardness of TFNP. In particular, it has been shown that hardness of TFNP hardness cannot be based on worst-case NP hardness, unless NP=coNP. Therefore, we ask the following question: What is the weakest assumption sufficient for showing hardness in TFNP? In this talk, I will answer this question and show that hard-on-average TFNP problems can be based on the weak assumption that there exists a hard-on-average language in NP. In particular, this includes the assumption of the existence of one-way functions. In terms of techniques, there is an interesting interplay between problems in TFNP and derandomization techniques. Based on joint works with Pavel Hubáček, and Moni Nao September 2012:00PM150 Forsyth Alina Ene, Boston University Submodular Maximization with Nearly-optimal Approximation and Adaptivity Abstract In this talk, we present recent progress on understanding the tradeoff between the approximation guarantee and adaptivity for submodular maximization. The adaptivity of an algorithm is the number of sequential rounds of queries it makes to the evaluation oracle of the function, where in every round the algorithm is allowed to make polynomially-many parallel queries. Adaptivity is an important consideration in settings where the objective function is estimated using samples and in applications where adaptivity is the main running time bottleneck. We present nearly-optimal algorithms for submodular maximization subject to a cardinality constraint and more generally, subject to multiple packing constraints. This talk is based on joint work with Huy Nguyen (Northeastern) and Adrian Vladu (Boston University). September 2712:00PM655 ISEC Lucianna Kiffer (internal) A Better Method to Analyze Blockchain Consistency Abstract The celebrated Nakamoto consensus protocol ushered in several new consensus applications including cryptocurrencies. A few recent works have analyzed important properties of blockchains, including most significantly, consistency, which is a guarantee that all honest parties output the same sequence of blocks throughout the execution of the protocol. To establish consistency, the prior analysis of Pass, Seeman and shelat required a careful counting of certain combinatorial events that was difficult to apply to variations of Nakamoto. Work by Garay, Kiayas, and Leonardas provides another method of analyzing the blockchain under the simplifying assumption that the network was synchronous. The contribution of this paper is the development of a simple Markov-chain based method for analyzing consistency properties of blockchain protocols. The method includes a formal way of stating strong concentration bounds as well as easy ways to concretely compute the bounds. We use our new method to answer a number of basic questions about consistency of blockchains: Our new analysis provides a tighter guarantee on the consistency property of Nakamoto’s protocol, including for parameter regimes which previous analysis could not consider We analyze a family of delaying attacks and extend them to other protocols; We analyze how long a participant should wait before considering a high-value transaction “confirmed”; We analyze the consistency of CliqueChain, a variation of the Chainweb system; We provide the first rigorous consistency analysis of GHOST and also analyze a folklore “balancing"-attack. In each case, we use our framework to experimentally analyze the consensus bounds for various network delay parameters and adversarial computing percentages. We hope our techniques enable authors of future blockchain proposals to provide a more rigorous analysis of their schemes. October 412:00PM150 Forsyth Maryam Aliakbarpour, MIT New directions in distribution testing Abstract In this talk, we will discuss new directions in testing properties of distributions. Distributions are at the core of data-related scientific endeavors, and due to their importance, they have been studied for past decades in Statistics and Computer Science. The goal is to find an algorithm to test the properties of distributions with provable performance guarantees, using as few samples as possible. We discuss the fundamental problems of identity (goodness of fit) and closeness (equivalence) testing concerning the new challenges: When the samples are sensitive data of participants, how can we make sure the output of our test preserves the privacy of the participants? We proposed algorithms to test uniformity, identity, and closeness of distributions that can achieve privacy with almost no cost. We demonstrate Our experimental evaluation. How can we test distribution in the presence of the noise? In what noise models can we perform better than almost linear? We consider the model that the underlying distribution is a mixture of the original distribution and the noise distribution. We proved if the noise distribution is known or if we can sample the noise distribution, we can test identity and closeness with almost no extra cost. October 1112:00PM150 Forsyth Tanay Mehta (internal) Resilience and the Coevolution of Interdependent Multiplex Networks Abstract We propose a new model for the study of resilience of coevolving multiplex scale-free networks. Our network model, called preferential interdependent networks, is a novel continuum over scale-free networks parameterized by their correlation ρ, 0 ≤ ρ ≤ 1. Our failure and recovery model ties the propensity of a node, both to fail and to assist in recovery, to its importance. We show, analytically, that our network model can achieve any γ,2 ≤ γ ≤ 3 for the exponent of the power law of the degree distribution; this is superior to existing multiplex models and allows us better fidelity in representing real-world networks. Our failure and recovery model is also a departure from the much studied cascading error model based on the giant component; it allows for surviving important nodes to send assistance to the damaged nodes to enable their recovery. This better reflects the reality of recovery in man-made networks such as social networks and infrastructure networks. Our main finding, based on simulations, is that resilient preferential interdependent networks are those in which the layers are neither completely correlated (ρ = 1) nor completely uncorrelated (ρ = 0) but instead semi-correlated (ρ ≈ 0.1 − 0.3). This finding is consistent with the real-world experience where complex man-made networks typically bounce back quickly from stress. In an attempt to explain our intriguing empirical discovery we present an argument for why semi-correlated multiplex networks can be the most resilient. Our argument can be seen as an explanation of plausibility or as an incomplete mathematical proof subject to certain technical conjectures that we make explicit. Joint work with Auroop Ganguly, Ravi Sundaram and Devesh Tiwari October 1812:00PM655 ISEC Manolis Zampetakis, MIT Iterative Methods for Efficient Statistics, in High Dimensions Abstract We consider the classical statistical problem of estimating to arbitrary accuracy the parameters of a multidimensional Gaussian distribution under the following real-world obstacles: (1) truncated samples, (2) inhomogeneous population. Statistical estimation from truncated samples is a classical problem, going back to Galton, Pearson, and Fisher. Truncated samples are samples that are only revealed if they fall in some subset S of the multi-dimensional Euclidean space. We provide and analyze an iterative algorithm with almost optimal sample and time complexity that recovers the parameters of a Gaussian distribution with arbitrary accuracy, from truncated samples. We prove the fast convergence of the celebrated iterative algorithm EM to efficiently estimate the means of Gaussian distributions in the inhomogeneous case of mixture of two Gaussian distributions. Abstracting the use of iterative algorithms in the previous important statistical problems, we consider the general question of analyzing the convergence of iterative algorithms. We prove that contraction maps provide a universal analysis tool for proving global convergence of iterative maps. Based on joint works with: Constantinos Daskalakis, Themis Gouleakis and Christos Tzamos. November 112:00PM655 ISEC Ran Cohen (internal) Must the Communication Graph of MPC Protocols be an Expander? Abstract Secure multiparty computation (MPC) on incomplete communication networks has been studied within two primary models: (1) Where a partial network is fixed a priori, and thus corruptions can occur dependent on its structure, and (2) Where edges in the communication graph are determined dynamically as part of the protocol. Whereas a rich literature has succeeded in mapping out the feasibility and limitations of graph structures supporting secure computation in the fixed-graph model (including strong classical lower bounds), these bounds do not apply in the latter dynamic-graph setting, which has recently seen exciting new results, but remains relatively unexplored. In this work, we initiate a similar foundational study of MPC within the dynamic-graph model. As a first step, we investigate the property of graph expansion. All existing protocols (implicitly or explicitly) yield communication graphs which are expanders, but it is not clear whether this is inherent. Our results consist of two types (for constant fraction of corruptions): Upper bounds: We demonstrate secure protocols whose induced communication graphs are not expander graphs, within a wide range of settings (computational, information theoretic, with low locality, even with low locality and adaptive security) each assuming some form of input-independent setup. Lower bounds: In the setting without setup and adaptive corruptions, we demonstrate that for certain functionalities, no protocol can maintain a non-expanding communication graph against all adversarial strategies. Our lower bound relies only on protocol correctness (not privacy), and requires a surprisingly delicate argument. More generally, we provide a formal framework for analyzing the evolving communication graph of MPC protocols, giving a starting point for studying the relation between secure computation and further, more general graph properties. November 812:00PM655 ISEC Seth Neel, University of Pennsylvania Heuristics for Differential Privacy Abstract We develop theory for using heuristics to solve computationally hard problems in differential privacy. Heuristic approaches have enjoyed tremendous success in machine learning, for which performance can be empirically evaluated. However, privacy guarantees cannot be evaluated empirically, and must be proven — without making heuristic assumptions. We show that learning problems over broad classes of functions — those that have polynomially sized universal identification sets — can be solved privately and efficiently, assuming the existence of a non-private oracle for solving the same problem. Our first algorithm yields a privacy guarantee that is contingent on the correctness of the oracle. We then give a reduction which applies to a class of heuristics which we call certifiable, which allows us to convert oracle-dependent privacy guarantees to worst-case privacy guarantee that hold even when the heuristic standing in for the oracle might fail in adversarial ways. Finally, we consider classes of functions for which both they and their dual classes have small universal identification sets. This includes most classes of simple boolean functions studied in the PAC learning literature, including conjunctions, disjunctions, parities, and discrete halfspaces. We show that there is an efficient algorithm for privately constructing synthetic data for any such class, given a non- private learning oracle. This in particular gives the first oracle-efficient algorithm for privately generating synthetic data for contingency tables. The most intriguing question left open by our work is whether or not every problem that can be solved differentially privately can be privately solved with an oracle-efficient algorithm. While we do not resolve this, we give a barrier result that suggests that any generic oracle-efficient reduction must fall outside of a natural class of algorithms (which includes the algorithms given in this paper). November 1512:00PM655 ISEC Albert Cheu (internal) Distributed Differential Privacy via Shuffling Abstract We consider the problem of designing scalable, robust protocols for computing statistics about sensitive data. Specifically, we look at how best to design differentially private protocols in a distributed setting, where each user holds a private datum. The literature has mostly considered two models: the “central” model, in which a trusted server collects users’ data in the clear, which allows greater accuracy; and the “local” model, in which users individually randomize their data, and need not trust the server, but accuracy is limited. Attempts to achieve the accuracy of the central model without a trusted server have so far focused on variants of cryptographic MPC, which limits scalability. In this paper, we initiate the analytic study of a shuffled model for distributed differentially private algorithms, which lies between the local and central models. This simple-to-implement model, a special case of the ESA framework of [Bittau et al., '17], augments the local model with an anonymous channel that randomly permutes a set of user-supplied messages. For sum queries, we show that this model provides the power of the central model while avoiding the need to trust a central server and the complexity of cryptographic secure function evaluation. More generally, we give evidence that the power of the shuffled model lies strictly between those of the central and local models: for a natural restriction of the model, we show that shuffled protocols for a widely studied selection problem require exponentially higher sample complexity than do central-model protocols. November 2912:00PM655 ISEC Ilias Zadik, MIT Revealing Network Structure, Confidentially Abstract Motivated by growing concerns over ensuring privacy on social networks, we develop new algorithms and impossibility results for fitting complex statistical models to network data subject to rigorous privacy guarantees. We consider the so-called node-differentially private algorithms, which compute information about a graph or network while provably revealing almost no information about the presence or absence of a particular node in the graph. We provide new algorithms for node-differentially private estimation for a popular and expressive family of network models: stochastic block models and their generalization, graphons. Our algorithms improve on prior work reducing their error quadratically and matching, in many regimes, the optimal nonprivate algorithm. We also show that for the simplest random graph models (G(n, p) and G(n, m)), nodeprivate algorithms can be qualitatively more accurate than for more complex models—converging at a rate of $1 / \epsilon^2 n^3$ instead of $1 / \epsilon^2 n^2$. This result uses a new extension lemma for differentially private algorithms that we hope will be broadly useful.


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Research Article

Journal of Agricultural, Life and Environmental Sciences. October 2020. 239-254
https://doi.org/10.22698/jales.20200020

# MAIN

• Introduction

• Data and Methods

•   Material

•   Conceptual structure and Methodology

•   Participation of smallholder farmers in Saving and Credit Cooperatives

•   Impact of SACCO on income of smallholder farmers

•   Bi-direction relationship of Income and participation in SACCO

• Result and Discussion

•   Descriptive statistics of smallholder farmers

•   Factors limits the participation of smallholder farmers in SACCOs

•   Descriptive statistics on attractiveness to SACCO’s products/services to participants

•   Statistics of model variables and definition of variables surveyed May-July 2019

•   Factors influence the participation of smallholder farmers in Saving and credit cooperatives.

•   The factors affect the income of smallholder farmers in Rwanda

•   Determination of bi-directional relationship between MSACCO and income

• Conclusion and Recommendation

Introduction

For past decades different studies revealed that access to finance is the shortest way to development of any country and whole world concern about ending poverty by opening up easy accessibility to finance for household, so poverty rate went on decreasing since past two decades (Rewilak, 2017). The promise of Saving and Credit cooperative is ability to reach out the poor household in both mobilizing and finance accessibility to them. SACCOs are playing big role in most of developing country in helping their members to increase income (Anania et al., 2015). and it is believed that microfinance enabled smallholder farmers to easily access to credit facilities without big collateral (Girabi et al., 2013). Saving and Credit Cooperatives (SACCOs) movement started in 2010 in different parts of Africa by establishing the law regulates these Microfinances. Some scholars confirmed that Saving and Credit cooperatives has been a good strategy for poverty reduction in some countries in Africa, according to the study done in Tanzania shows that Saving and Credit cooperatives is key among the factors for poverty reduction reduction (Kihwele et al., 2015).

In Rwanda, Saving and Credit Cooperatives (SACCOs) are main provider of finance to smallholder farmers in rural areas (Nair et al., 2018). As at December 2016, out of the 472 microfinance institutions operating in Rwanda, 17 are microfinance institutions with limited liability company status; 455 are Savings and Credits Cooperatives (SACCOs): These include 416 Umurenge Saving and Credit Cooperative (U-SACCO) and 39 non U-SACCOs (Bank et al., 2017). The concept of Umurenge Savings and Credit Cooperatives (U-SACCOs) was based on an understanding that banks and other financial institutions were more concentrated in urban areas whilst the majority of the Rwandan population lives in rural areas and totally excluded from the formal financial institutions. Umurenge SACCOs are Sector level based Saving and Credit Cooperatives (one SACCO per sector level) and were established in 2008 with the aim to boost up rural savings and provide Rwandans with loans to improve their earnings and enhance their livelihoods. This program has contributed financial service in Rwanda, The Fig. 1.demonstrates percentage savings of members in U-SACCO each year, since 2010 to August 2018. Deposits of members in Saving and Credit cooperatives (SACCO) is over 67.3 billion Rwandan francs where 95% of this deposit was saved in U-SACCOs and 5% in other SACCOs.

##### Fig. 1.

Deposit collected in U-SACCO between2010 and August 2018 (Rwanda).

Data and Methods

Material

The study adopted a cross-sectional design and data were collected at a single point in time. This design was preferred, for despite its simplicity, it allows one to collect data about past and current experiences to identify possible associations (Kwai and Urassa, 2015). This study was conducted in BUSASAMANA Sector, Nyanza district, southern province of Rwanda between May and July 2019, Busasamana Sector in Located in Nyanza District in southern province of Rwanda. Geographically has area of 49.3 km2 and population of 42,870. The big part of this sector is rural area and main economic activities in this sector are substance agriculture and trading, most farming is done on small scale and main agriculture produce are rice, maize, beans, banana plant, peas, vegetables, Mushroom and coffee on small scale. The study used a cross sectional survey design where the purposive sampling was used to select 200 smallholder farmers where 124 are members of SACCO and 76 are non-members. The survey conducted by a numerator and questionnaires were designed for face-to-face interview. The data was analyzed by using Stata14 statistical software. Sample size is appropriate if it enables the researcher to make an unequivocal judgment that a statistical result is correct to a specified meaningful effect (Malone et al, 2016). This study used 200 sample sizes which was calculated and Since the proportion of the population was not known, the study assumed P=0.9, (1-P) = 1-0.9= 0.1, Z = 1.96 and d is assumed to 0.05. The results came out as 138.3 but the researcher used 200 sample size to have more accurate results, according to the study, the larger the sample, the smaller the margin of uncertainty (confidence interval) around the results (Conroy, 2020).

Conceptual structure and Methodology

This study used three models in responding to the objectives of the study. Linear regression Model was used to determine the effect of SACCO on income of small holder farmers, Logistic model to examine the factors that influences smallholder farmers’ participation in SACCO (Damodar, 2004). Seemingly unrelated regression (SUR) to analyses bidirectional relationship of participation in SACCO and income. Nine variables were used to in each empirical model to achieve the objective of this study. This study used three (3) econometric models in order to empirically achieve the objectives and 8 independent variables1) were employed.

1) Independent variables: X1=Age of participant, X2=Sex, X3=marital status, X4=Household size, X5=Education level, X6=Income source, X7=Requirements to join SACCOs, X8=Income.

Participation of smallholder farmers in Saving and Credit Cooperatives

The choice to participate in SACCO or not participate in SACCO can affect income of smallholder farmers. This ends up as bi-outcome dependent variables, dependent variables (MSACCO) is participation of smallholder farmers has two (2) outcome, probabilities of smallholder farmers to be a member and probabilities of smallholders farmers to be non-member of Saving and Credit cooperative (SACCO) and those variables are code as binary 1(one) for member and 0 (Zero) as non-member of Saving and Credit Cooperative. Logistic regression analysis model for dichotomous outcomes showed how the logistic regression is more important than other linear regression more when outcome are binary because mathematically is extremely flexible and easily used function and it has theoretical meaningful interpretation (Afolami et al., 2015). This study used logistic regression model to examine the participation decision of smallholder farmers in Saving and Credit Cooperative.

Binary logistic regression model does not make assumption about the distribution of the independent variables (Afolami et al., 2015). The relationship between the predictor and response variables is not a linear function in logistic regression; instead, the logistic regression function is used, which is the logit transformation of p:

 $$L=Log\left[\frac{P(x)}{1-P(x)}\right]=\beta_0+\beta_1X_1+\beta_2X_2+\cdots\;+\beta_nX_n$$ (1)

L stands for logistic model, P(x) is a probability of event to occur, is given by (1), then the probability of not event occur is 1－P(x) and then P(x) / (1－P(x)) is simply the odds ratio in favor of event to occur. And notations of is intercept regression is ${\beta }_{0}$, and, ${\beta }_{1},{\beta }_{2},\dots +{\beta }_{n}$ are coefficient of independent variables and X1, >X2…Xn are independent variables of regression. The ratio of the likelihood of occurrence of an event to the probability of non-occurrence of the event as well the outcome of logit model and Since logistic regression calculates the probability of success over the probability of failure, the results of the analysis are in the form of odds ratio. Logistic regression also provides knowledge of the relationships and strengths among the variables. This Logistic formula can be used to estimate the change in probability of smallholder farmers who participated in Saving and Credit Cooperative (SACCO) and comparison with non- participation in SACCO provides the measurement the impact to smallholder farmers at end. Decision of smallholder farmer to participate in SACCOs (being a member of SACCO) is assumed to help them to earn more income and the below equation declares the variables employed.

 $$MSACCO(0,1)=\beta_0+\beta_1age+\beta_2sex+\beta_3MRTSTATUS+\beta_4HSHDSIDE\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\beta_5EDUCLEVEL+\beta_6incomesorc+\beta_7RQMTS+\beta_8income+\epsilon$$ (2)

Impact of SACCO on income of smallholder farmers

This study also used linear regression model (OLS) to analyze the effect of the saving and Credit Cooperative in on income. Ordinary least-squares (OLS) regression is a generalized linear modeling technique that may be used to model a single response variable (dependent variable) and may be applied to single and well as multiple explanatory variables and also categorical explanatory variables that have been appropriately coded, this study adopted this model on analysis of impact of SACCO on income as single response dependent variable with multiple independent variables (Hutcheson, 2011). This model assumes there is linear relationship between independent variable and dependent variables (Hutcheson, 2011). Ordinary least squares model is written as below

 $$Y=\beta_0+\beta_1X_{1J}+\beta_2X_{2J}+\cdots\;+\beta_KX_{KJ}+\epsilon$$ (3)

The parameters βj, j=0, 1, 2……k are known as regression coefficients. This model describes a hyperplane in the k – dimensional space of the independent variables Xj’s. The parameter βj represents the expected change in the dependent variable Y which is income in this study per unit change in Xj , when all of the remaining predicted variables are constant. This study adopts this model to estimate the change of dependent variable Y (income) caused the change in participation in SACCOs as independent variable holding other variable constant.

 $$INCOME=\beta_0+\beta_1age+\beta_2sex+\beta_3MRTSTATUS+\beta_4HSHDSIZE\\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\beta_5EDUCLEVE+\beta_6INCOMESORC+\beta_7RQMTS+\beta_8MSACCO+\epsilon$$ (4)

Bi-direction relationship of Income and participation in SACCO

This study used SUR model find out bi-directional relationship of income and participation in (MSACCO) variables as this study assumed two variables are independent to each other. Simultaneously modeling of seemingly unrelated model is appropriate estimated two equations (Mujuru et al., 2020). Two variables are examined, Participation in SACCO and income are expected to have a bi-directional relationship and hence the need to estimate the two equations and the SUR model is supposed to presented as following (Zellner, 1962). We consider here a model comprising of M multiple regression equations of the form

 $$y_{ir}=\beta_iX_i+\epsilon_{ir},\;\;\;i=1,2,\;\dots,M$$ (5)

Here i represents the equation number, r=1,..,R is the time period. The number of observations R is assumed to be large, so that in the analysis we take R→∞, whereas the number of equations M remains fixed.

Each equation i has a single response variable y_(ir) and a k_(i)- dimensional vector of regressors X_(ir). If we stack observations corresponding to the ith equation into R -dimensional vectors and matrices, then the model can be written in vector form as

Where yiand ${ϵ}_{i}$ are (R × 1) vectors, Xi is a (R × ki) matrix, and ${\beta }_{i}$ is a ki × 1 vector.

Finally, if we stack these m vector equations on top of each other, the system will take the form

 $$\begin{pmatrix}y_1\\y_2\\\vdots\\y_M\end{pmatrix}=\begin{pmatrix}X_1&0\cdots&0\\0&X_2\cdots&0\\\vdots&\vdots&\\0&0&X_M\end{pmatrix}\begin{pmatrix}\beta_1\\\beta_2\\\vdots\\\beta_M\end{pmatrix}+\begin{pmatrix}\varepsilon_1\\\varepsilon_2\\\vdots\\\varepsilon_M\end{pmatrix}=\mathrm X\beta+\varepsilon.$$ (6)

The assumption of the model is that error terms are independent across time, but may have cross-equation contemporaneous correlations. Thus we assume that

 $$E\lbrack\epsilon_{ir}\;\;\;\epsilon_{is}\;\;\;\left|X\right.\rbrack=0$$ (7)

Basing on equation (5) this study M is considered to be 2(M=2), where Y1 Participation in SACCOs (MSACCO), Y2 is income of respondents (smallholder farmers) and X1 → X8 are explanatory variables for each equation. This model was used to analyze simultaneous effects of income and membership (Participation in saving and credit cooperative) more specific to check the bi-directional relationship for both income and participation in SACCO (MSACCO). SUR was capable in dealing with the correlation of error term between in Income and MSACCO among two multiple regressions.

This study has variables, which affect both regressors (Income and MSACCO) and also regressors (Income and MSACCO) are allowed to be endogenous variables to each other and previous study used SUR model to investigate bidirectional relationship between export and growth of developing countries (Kónya, 2006). This study generalized the two multiple equation into simultaneous equations model SUR, assuming there is correlation in both variable influence each other. Some identified factors used in estimating Income are Age, Sex, Marital status, Household size, education level, require to participate in SACCOs (RQMTS) and MSACCO. While MSACCO are Age, Sex, Marital status, Household size, education level, RQMTS (Requirement to participate in SACCO), Income source, and Income. A Seemingly Unrelated Regression Model comprises several individual relationships that are linked by the fact that their disturbances are correlated of which researcher assumed the correction of disturbance as mentioned above, in this study the simultaneity effect was acknowledged between Income and MSACCO (participation in SACCO) which were recognized as Endogenous variables.

Result and Discussion

Descriptive statistics of smallholder farmers

Table 1 presents the descriptive statistics of variables which describes main variables about smallholder farmers surveyed in Busasamana sector in south province of Rwanda in 2019. Among sample size of 200 of smallholder farmers, analysis demonstrates that 124 smallholder farmers of participated in SACCOs and 76 smallholder farmers are non-participants, most frequent gender of smallholder farmers participated in SACCOs is female (n=109, 54%) in this survey. Majority of smallholder farmers are aged between 22 and 40 years old (n=117,58.5%), and majority of smallholder farmers are married (n=174, 87%). Of total sample size, the most frequent smallholder farmers’ household size is made up of between one person to five people (n= 114,57%). The study revealed that most of Rwandans who engage in smallholder farming are of primary level education with most frequent in this survey (n=137, 68.5%) and primarily income source is from agricultural production with most frequent of n=178( 89%)of smallholder farmer surveyed as well most frequent grouped income earning is less than or equal to 20000 Rwandan francs per month(n=138, 69%), it was also revealed that most of farmers know(aware) about requirements to participate2) in SACCOs with most frequent in this survey (n= 163, 81.5%) of which demonstrate 18.5% farmers may be limited to participate in SACCOs because they don’t know about requirements.

2) The requirements to participate in SACCOs are 1) paying membership fee, 2) attending the meeting, 3) paying monthly fee, and 4) meeting quantity of output.

#### Descriptive statistics of sampled smallholder farmers in Rwanda

 Variables Frequency(n) Percentage (%) MSACCO Member =1 124 62.0 Non-member=0 76 38.0 Total 200 100 Age 22-40 117 58.5 >40-60 74 37.0 >60-84 9 4.5 Total 200 100 Sex Male=0 91 45.5 Female=1 109 54.5 Total 200 100 Marital status Single (widow, divorced)=0 26 13.0 Married=1 174 87.0 Total 200 100 Household size 1-5 number 114 57.0 <5－10 number 82 41.0 <10 4 2.0 Total 200 100 Education level (academic years) Non=0 15 7.5 Primary=6 137 68.5 Secondary=12 34 17.0 Bachelor=16 2 1.0 Masters=18 12 6.0 Total 200 100 Requirement to participate in SACCO No (No and Don't know)=0 37 18.5 Yes =1 163 81.5 Total 200 100 Income source Off farm = 0 22 11.0 Farm production =1 178 89.0 Total 200 100 Income in Rwandan francs (RWF) <=20000 138 69.0 >20000－40000 44 22.0 40000<80000 10 5.0 >80000 8 4.0 Total 200 100

Source: field survey, 2019.

Factors limits the participation of smallholder farmers in SACCOs

The results from Table 2,demonstrates that majority of smallholder farmers who doesn’t participate in saving and credit cooperative (SACCO) because of distance from smallholder farmer’s house to the SACCO which counts (N=87, 43.5%) which followed by second highest frequency of smallholder farmers claimed of not participated because of lack of information on products/ services of SACCO (N=41, 20.5 %), this study also revealed that smallholder farmers do not participate in SACCO because SACCOs in Rwanda are non- automated in terms of transactions (N=34, 17%), and least smallholder farmers found out that did participate in SACCO due high membership fee, small loan issued by SACCO and due high interest rate with least frequency of N=18 (9%), N=8 (4%) and N=3 (1.5%) respectively.

#### Statistics on limitation on participants in SACCOs

 Reasons not join Frequency(N) Percentage% Non automated 34 17.0 Distance 87 43.5 Small loan 8 4.0 No information 41 20.5 High interest rate 3 1.5 High membership fee 18 9.0

Source: Field survey, 2019.

Descriptive statistics on attractiveness to SACCO’s products/services to participants

The results in Table 3 demonstrate statistics 12.5% of smallholder farmers received loan from SACCO and 87.5 % didn’t receive loan and concern the use of accounts, majority of participants in hold current accounts which count of 73 % of member of SACCO and 27 % of participant hold long-term/saving account.

#### Statistics on attractiveness of products provided to participants by SACCOs

 Service provided Frequency Percentage Received loan 25 12.5 Not received loan 175 87.5 Current account user 146 73.0 long-term account user 54 27.0

Source: field survey, 2019.

Statistics of model variables and definition of variables surveyed May-July 2019

Variables provided in the Table 4 are summary statistics of variables used in analysis; nine variables (9), 200 observations for each variable were sampled. According to survey average of 0.62 smallholder farmers participated in SACCO (MSACCO); highest number of smallholder farmers are female with average 0.545 and most of them are married with average 0.87 married farmers; average income of smallholder farmer in Busasamana sector is 22725 FRW, higher earned 195000 and lowest earn 10000 FRW and main income source for most farmers is farmer production with average 0f 0.89. Average education level of smallholder approximate 7 years (primary level with first year of secondary level), smallholder with highest level is 18 years which equivalent to Master’s degree and lowest education level is 0 (un school) farmers; average age of smallholder farmers approximately 41 year old, youngest is 22 year old and oldest is 84 years old and size of highest household size of this sample is consist of 11 people and lowest size is made of one person with average size 5 person per household. This survey also demonstrate the average smallholders know about requirements to participate (RQMTS) in SACCO to 0.815.

#### Summary statistics of model variables surveyed between May-July 2019

 Variables Units Obs Mean Standard deviation Min Max MSACCO (Member=1,Nonmember=0) Number 200 0.62 0.49 0 1 Sex (Female=1,Male=0) Number 200 0.55 0.50 0 1 Marital status (Married =1Single=0) Number 200 0.87 0.34 0 1 RQMTS (Yes=1,No=0) Number 200 0.82 0.39 0 1 Income source (Farm production=1,Off farm=0) Number 200 0.89 0.31 0 1 Income Ten thousand FRW 200 2.29 2.56 1 19.5 Education Years 200 7.39 4.08 0 18 Age Years 200 40.90 11.42 22 84 Household size Number 200 5.29 1.95 1 11

Source: field survey, 2019.

Factors influence the participation of smallholder farmers in Saving and credit cooperatives.

From data analysis in response to second of objective of this study, logistic regression model was used and the outputs are shown in the Table 5 revealed the results on factors influence the participation of smallholder farmers in Saving and credit cooperative (SACCO). The findings show variables EDUCLEVEL (education level), RQMTS (requirements) and income are all significant variables at p < 0.05, p < 0.01, p < 0.001. The interpretations of the results are following:

#### Results of logistic model on factors that influence smallholder farmer participation in SACCOs

 variables coefficient Std.err. Z-values P-value Constant -1.02 0.08 -0.72 0.47 Age -0.03 0.02 -1.42 0.16 Sex -0.12 0.38 -0.37 0.75 Marital status 0.06 0.59 0.11 0.91 Education 0.14* 0.05 2.63 0.009 Household size -0.18 0.09 -1.81 0.07 RQMTS 4.99*** 0.86 5.25 0.000 Income 0.23* 0.09 2.56 0.01 Income source 0.25 0.61 0.42 0.68

Note: *, **, *** Indicate significance 10%, 5% and 1% respectively. Source: field survey, 2019.

Coefficient of Education level of smallholder farmers is significant at p < 0.01 and has negative influence on decision taken by smallholder farmer to participate in SACCO. This implies that smallholder farmers acquire higher education level, the likelihood of participating in SACCO tend to decreases. This is because, this study is limited to smallholder farmers , according to recent study revealed, majority of people in agriculture activities did not acquire formal education because those who obtain knowledge (education) participate less in rural agricultural activities since they look for other opportunities out of farming, this ends to less educated people in farming and this is the main reason to which this study shows that high educated farmers are less likely to be a member of SACCO. This study focused only on smallholder farmers and since it is specific on smallholder farmers, it clearly indicates that as high education is acquired less participation in Saving and credit cooperative (SACCO).

The results found out that coefficient of requirement to join SACCO (RQMTS)is positive and strongly significant influence smallholder farmers on being the member of SACCO. The positive outcome implies as more requirements put in place to join saving and credit cooperative there is like hood of farmers to participate in SACCOs. This point out that more requirements more participation in SACCOs, this because people put their money in financial institution where they trust requirements.

The results on coefficient of Income revealed that income at p < 0.05 is strongly positive and significant influence smallholder farmers to participate in SACCO. This positive result means, for smallholder farmers who earn more income are more likely to participate in SACCO, because they are looking for security of their net income as well as to benefit from savings. This result is backed up by other previous study which examined the determinants of participation of household in SACCO in Uganda, their results indicated that the likelihood of membership in SACCOs and depositing in a SACCO increased with growth in incomes of household (Mpiira et al., 2013). It implies that earning income affect the decision taken by smallholder farmers to be member or not member in positive association.

The factors affect the income of smallholder farmers in Rwanda

The findings out of 200 observations reveals that 5 variables (Household size, Education level, requirement, MSACCO, income source) are all significant at *p < 0.05, **p < 0.01, ***p < 0.001. The interpretations below for predictor variables provide information on associated dependent variable (income) based on outcome shown in Table 6. Household size for smallholder farmer is statistically significant at p < 0.01 and positively influence on income of smallholder farmers. The positive relationship means increase household size smallholder leads to large number of labor for farming which results in high production and hence increase of income of household. The outcome of this study shows importance of household size in income generation strategies of smallholder farmers. This result is in line with outcome previous which confirmed that annual income is positive and significant associated number of household (Ngongi et al., 2014).

#### Results of ordinary least squares on factors affecting smallholder farmers' income

 Variables Coefficient Std.Err. T-Stat Constant 0.02 1.23 0.02 Age 0.02 0.02 1.23 Sex 0.54 0.36 1.49 Marital Status 0.45 0.57 0.79 Household size 0.26** 0.09 2.63 Education level 0.21*** 0.04 4.35 RQMTS -1.22* 0.55 -2.19 MSACCO 1.02* 0.44 2.30 Income source -2.04*** 0.59 -3.42

Note: *, **, *** Indicate significance 10%, 5% and 1% respectively.

Coefficient of education level is positive and significant at p < 0.001 influences on income of smallholder farmer in Rwanda. The positive output of education level demonstrates that smallholder farmers who acquire more education level access on more information on farming technics and which help them to increase productivity as well as access to market for income increment. This indicates the importance of education of smallholder farmers for income generation, which result in acquired knowledge in agricultural production and market access.

Coefficient of RQMTS (requirement to join SACCO) is negative and significant at p < 0.05 influences on current income. Negative coefficient indicates that more requirements put in place to join SACCO tends to decrease the current net income of participants, because for more requirements requires more money to pay which decrease current net income (take home income).

MSACCO (membership of SACCO) is significant at 5% and strongly positive influence on income. The positive influence implies that smallholder farmers who are members of Saving and credit Cooperatives (SACCO) tend to have higher income than non-members. Smallholder farmers who are members of SACCO access to finance easily and invest in farming technology for high production as well as help in accessing to market for their productivity to direction of more income generation.

Income source Coefficient is negative and significant at p < 0.001 influencing on income. This indicates with fact that smallholder farmers with more income source tends to have less net income. This study is backed up by previous study, according to relationship between non-farm income and overall income is U-shape because for long-term investments and barriers to enter into market which tends to decrease overall income of farmer. This demonstrates importance of smallholder farming giving more time and high technology in farm production can generate more income.

Determination of bi-directional relationship between MSACCO and income

Seemingly unrelated regression model (SUR) was used to examine whether participate in saving and credit cooperative (SACCO) and income influence each other. This model assumes that both income and MSACC are Endogenous variables. All the rest of variables are exogenous and are same for both equations. Variables used are Age, sex, marital status, household size, education level, requirements, income, income source, MSACCO. The outcomes of seemingly unrelated regression are shown below:

In Table 7 shows correlation between two estimates equation was tested by Breusch-Pagan test of independence and is significant because P is very small (p=0.020) which implies there is correlation between across two equations.

#### Results of SUR model on determination of bi-directional relationship between MSACCO and income

 Variable MSACCO INCOME Coeff. Std.err. Z-score Coeff. Std.err. Z-score Age -0.004 0.003 -1.46 0.024 0.017 1.46 Sex -0.037 0.058 0.64 0.555 0.358 1.55 Mrtstatus 0.003 0.091 0.03 0.426 0.563 0.76 Hshldsize -0.033* 0.016 -2.11 0.276** 0.095 2.90 Educlevel -0.024** 0.007 -3.13 0.221*** 0.046 4.74 RQMTS 0.717*** 0.073 9.77 -1.887*** 0.544 -3.49 Income 0.052*** 0.011 4.65 - - MSACCO 1.987*** 0.427 4.65 Income source 0.089 0.096 0.95 -2.022 0.582 -3.48 Constant 0.192*** 0.384 2.00 -0.372*** 1.203 -0.31

Breusch-Pagan test of independence: chi2 (1) = 5.410, Pr = 0.020

Note: *, **, *** Indicate significance 10%, 5% and 1% respectively.

Coefficients of both MSACCO and Income are significant at p < 0.001 and strongly positive influence each other. This implies that smallholder who earner more income tends to participate in Saving and Credit Cooperatives (SACCO) and as well as smallholder farmers who participate in SACCO earn more income than non-members.

Coefficient of number of household of smallholder farmers is negative influence and significant at p<0.05 to MSACCO, negative sign implies increase smallholder farmers with higher number of household tend to less likely participate in SACCO, the results with Logistic is household size is not significant, where as in SUR model household size is significant. Coefficient of household size to income is positive and significant at p < 0.01 to income which implies that smallholder with bigger number of household tend to have higher income which means more household number may have high income earning since individual member contribute to income on household.

Education level of smallholder of farmer has negative influence to MSACCO and significant at p < 0.001, this indicates that for more educated smallholder tends to less participate in SACCO, this is due to more educated people tend to migrate from rural areas which means most of smallholder farmers are less educated. In contrast, Coefficient of Education level of smallholder has positive influence on income and significant at p < 0.001, positive sign implies that more educated smallholder farmers tends to earn more income, because more educated people always access to information where to invest and earn more. The results of education level variable from SUR Model are better than running regression individually; coefficients are bigger in SUR model results compared to individual equation.

Coefficient of requirements to join SACCO (RQMTS) to MSACCO is positive influenced and significant at p < 0.001. This positive sign means as more requirements put in place to participate in SACCO, smallholder farmers tend to participate in SACCO. This explained by the fact that most people put trust/money more where they can have many agreements/contract with financial institution, for any financial requirement is followed with certain contract. However, Coefficient of requirements to join SACCO (RQMTS) to income is negative influenced and significant at p < 0.001. This implies as more requirements put in place net income of smallholder farmers decrease, since requirement cost more that why more requirement less net income.

Income source Coefficient is negative and significant at p < 0.001 influencing on income. This indicates with fact that smallholder farmers with more income source tends to have less net income. This study is backed up by previous study, relationship between non-farm income and overall income is U-shape because for long-term investments and barriers to enter into market which tends to decrease overall income of farmer. This demonstrates importance of smallholder farming giving more time and high technology in farm production can generate more income (Rapsomanikis, 2015).

Conclusion and Recommendation

This paper is purposely to assess the impact of SACCO on income of smallholder farmers. Primary data was collected and analyzed with help of 3 models, both OLS and SUR results demonstrated the impact of SACCO on income but OLS results may have biasness because of endogeneity between MSACCO (participation in SACCO) and income. The study also revealed that standard errors of SUR estimator are lowest and highest precision of estimate compared to OLS. SUR estimates are consistent compared to OLS estimates because SUR takes in account the correction in error terms. (Cadavez and Henningsen, 2012). Basing on the results from SUR estimator, This study found that participation in Saving and credit cooperatives (SACCO) has impact on income of smallholder farmers in Rwanda, there is increasing in income earning of members than non-members of SACCOs and the finding of this study is backed up by previous studies revealed the impact of Saving and credit cooperative on economic development and wellbeing of members/ households in Rwanda but those studies remained limited at well-being household in general. In addition the study found out that females are mostly engage in smallholder farming. Furthermore, findings of this study also revealed that income of smallholder farmers are significantly explained by other factors such as education level of smallholder farmers and most of farmers are primary level educative, Requirements put in place to Join SACCO, household size of smallholders of which most farmers are aware of requirement to participate in SACCO and also income source of it was found (Jonathan and Marthe, 2018).

Study tacked on factors influence decision of smallholder farmers to participate in Saving and credit cooperative (SACCOs) and results found that factors such as Education level, income and Requirements put in place to Join SACCO to be significantly affecting the decision taken by smallholder farmers to engage in SACCOs.

The finding also demonstrates that there is bi-direction effect between Income and participation in saving and credit cooperative (SACCO), smallholder farmers who earn more participates in SACCOs for security of their money and as well as participation in saving and credit cooperative boost income earning due to investment done out of loan and interest from savings. The study also tacked how members of SACCO grabs the services provided, it was found that it is small percentage of smallholder farmers request for loans, only 12.5% received loan and most of smallholder farmers hold current account compared to saving/long-term account holders which counts 73 percent and 27 percent respectively. However, the findings of this study disclosed the factors hinders smallholder farmers to participate in Saving and Credit cooperative, main factors revealed are two, long distance from SACCO to resident of some smallholder farmers and lack of information on some smallholder farmers on saving and credit cooperative’ operations. For previous studies urged that contributed much to accessibility finance in Rwanda of which every Umurenge SACCO is located in 5 KM radius to Rwandan population. However, this study revealed that 43.5% of non-members claimed of long distance from their resident, which implies some smallholder farmers whom are not located in 5KM radius. Finally, this study came up with following recommendations.

Government should motivate more farmers to participate in SACCOs, this study found out that members of SACCOs earn more than non-members since income generation for smallholder farmers. Motivating by put incentives to join can boost income generation of farmers as well economy of the country as it should be shortest way to end poverty in all corners of the country.

Government should create off farm activity for smallholder farmers for added income earning since this study found that there is bi-direction relationship in participation in SACCO with income of smallholder farmers which with can lead to increase financial inclusion as well as more income generation which may result from interest gained for long-term savings.

Both Rwanda cooperative Agency and National Bank of Rwanda as regulators of SACCOs should access SACCOs located far from smallholder farmers and establish more branches for ease of accessibility to SACCO.

Government should put effort in automating SACCOs to attract more members to participate and for easy access to the product/service of SACCOs where ever in Rwanda.

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