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Laura Chioda World Bank. Sampling and Statistical Power. Outline. Ingredients of a general recipe called “Statistical Sampling Theory” Key message : a successful design involves some guess work. It is important to have a general rule, but then needs discussion depending on the case at hand

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## Sampling and Statistical Power

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**Laura Chioda World Bank**Sampling and Statistical Power**Outline**• Ingredients of a general recipe called “Statistical Sampling Theory” • Key message: a successful design involves some guess work. It is important to have a general rule, but then needs discussion depending on the case at hand • Use randomization as a benchmark • I will not deal with non-experimental designs in general, though what follows can be straightforwardly adapted to cover that case**Basic set up (1)**• Key ingredient for impact evaluation: • Questions, what do we want to learn about this initiative/project? • How do I identify the effect that answers my questions? After this morning possibly a bit more convinced that experimental methods can be ethical, interesting and funny!! • What is Next? DATA COLLECTION, sampling and sampling design • Set up: assume we have performed our lottery and we have identified our treatment and the control groups • What we would like to measure is difference: Effect of the Program = Mean in treatment - Mean in control Example: average farmers' income who adopted fertilizer because of new incentive program vs the farmers in the control group who didn’t receive any incentives**Basic set up (2)**• Why is Randomization good? • may produce experimental groups that differ by chance, however. These differences are random errors, not biases • when done properly allows us to identify casual effects attributable to the program. • Bottom line: randomization removes bias, but it does not remove random noise in the data. • The latter statement is the reason why we worry about sampling!!!**Basic set up (3)**• In a world with no budget constraint we could collect data on ALL the individuals (universe) in the treatment and in the control groups. • In practice, we do not observe the entire population, just a sample. Example: we do not have data for all farmers of the country/region, but just for a random sample of them in treatment and control groups • Hence we do not measure the true effect, but we estimate the mean outcome of interest by computing the average in the sample. Example: we compute the average income for farmers in the treatment vs avg. income for farmers in the control group. Bottom line: Estimated Effect = True Effect + Noise**Planning Sample Size for Randomized Evaluations**• The noise coming from the fact we collect data for a subset (sample) of the entire universe forces us to think about sampling & how to minimize the prob. that we may end up with a wrong conclusion (scary!!! Not really) • Question: How large does the sample need to be to credibly detect a given effect size? • What does “credibly” mean here? It means that I can measure with a certain degree of confidence the difference between participants and non-participants • Key ingredients: • number of units (e.g. villages) randomized • number of individuals (e.g. households) within units • info on the outcome of interest and the expected size of the effect on this outcome**Let’s go back to basics:**The general idea can be easily put across by testing the hypothesis that the effect size is equal to zero We want to test: Against: >> Can be done for different groups of individuals Hypothesis Testing (1)**Hypothesis Testing (2)**It is reasonable to think of the following “ideal” property of any testing procedure: minimize disappointment , but allow for a minimum degree of error TrueEffectSize**Two Types of Mistakes (1)**First type of error: conclude that the program has an effect, when in fact at best it has no effect • Significance level of a test: Probability that you will falsely conclude that the program has an effect, when in fact it does not • If you find an effect with a level of 5%, you can be 95% confident in the validity of your conclusion that the program had an effect • For policy purpose, you want to be very confident of the answer you give: the level will be set fairly low • Common levels are: 5%, 10%, 1%**Two Types of Mistakes (2)**Second Type of Error: You conclude that the program has no effect when indeed it had an effect, but it was not measure with enough precision (sort of technical, see next) • Power of a test: Probability to find a significant effect if there truly is an effect • higher power is better since I am more likely to have an effect to report • don’t think about statistics, simply look at the graph! **Practical Steps**• Set a pre-specified confidence level (5%) – i.e. just set the initial point of the line in the graph • Decide for a sample size that allows to achieve a given power. • Common values used are 80% or 90%. Intuitively, the larger the sample, the larger the power. • Power is a planning tool: one minus the power is the probability to be disappointed…. • Set a range of pre-specified effect sizes (what you think the program will do) • What is the smallest effect that should prompt a policy response? Aka minimum detectable effect**Picking an Effect Size**• Wait a minute is this a catch 22, we do not know the true effect? No, it is a thought experiment, we gain insight from • economics • past data on the outcome of interest or even past evaluations • What is the smallest effect that should justify the program to be adopted? • Cost of this program v the benefits it brings • Cost of this program v the alternative use of the money • Common danger: picking effect size that are too optimistic—the sample size may be set too low to detect an actual effect **Practical Steps and “magic formulas”**• Proposition I: There exists at least one statistician in the world who has already put into a magic formula the optimal sample size required to address this problem • Proposition II: The rule has also been implemented for almost all computer software • Not difficult to do, and only requires simple calculations to understand the logic (really simple!)**Picking an Effect Size**General “rule”: the sample size required is a function of: • Significance level (often set to 5%) • Minimum detectable effect – you set this • Power to detect it (often set to 80%) • Variance of the outcome of interest before the intervention takes place (derived from baseline data)**The Design Factors that Influence Power**• Which elements can influence the power? • The level of randomization • Stratification • Availability of Control Variables • Very fancy words, to express very simple concepts… • Statistics is nothing more than common sense**Level of RandomizationClustered Design (1)**Cluster (or group) randomized trials are experiments in which social units or clusters rather than individuals are randomly allocated to the intervention group Examples: • First randomize villages • Then observe outcome variables at the household level • In an education program, randomize schools • Then look at students’ achievement**Level of RandomizationClustered Design (2)**• Cluster randomization provides unbiased estimates of intervention effects for the same reasons that individual randomization does… • However, makes our lives a bit more complicated when we have to do our power calculations: • the statistical power or precision of cluster randomization is less than that for individual randomization, and often by a lot! • Necessitates larger samples to increase statistical power!**Implications of Clustering**• Forced to “cluster”, i.e. to take into consideration that there are some elements that affect all the people in a given village/group/ basin in the same way. • Why? The outcomes for all the individuals within a cluster may be correlated • All villagers are exposed to the same NGO • The members of a village interact with each other sharing knowledge, technology and best practices.. • Individuals in a given area are exposed to the same soil characteristics. • The sample size needs to be adjusted for this correlation • The more correlation between the outcomes, the more we need to adjust the standard errors**In Practice…**• It is extremely important to randomize an adequate number of clusters. • The general result is that the number of individuals within clusters matters less than the number of clusters • Think that the “law of large number” applies only when the number of clusters that are randomized is sufficient • Number of Clusters == Number of Observations**Stratification (1)**• What do we mean by stratification of the sample and what is a stratum? • Esoteric expression to simply say: the more elaborate is the question you want to answer the larger is the requirement in terms of sample size • Compare the 2 scenarios: • Research question 1: I would like to know the average effect of the fertilizer program on farmers’ income? • Research question 2: I would like to know the average effect of the program for female headed low income families under the age of 35 , living in the north east? • Question: keeping all other elements constants (minimum size effect, power target, significance level) which research question will need more data in order to be answered with precision?**Stratification (2)**• To improve precision one could block or stratify experimental sample members by some combination of their baseline characteristics, and then randomize within each block or stratum • Factors used for blocking in social research typically include: • geographic location • demographic characteristics • past outcomes Punch Line: stratification demands for larger sample size, but we get more detailed answers without sacrificing precision**Control Variables (1)**• Good news: if control variables such as • Gender, education levels, age (demographics) • Farms characteristics, Soil composition Are available • Then the precision of our estimates will increase hence the sample size requirement decreases. This reduces variance for two reasons: • Reduces the variance of the outcome of interest in each stratum, and • Reduces the correlation of units within clusters • Warning: control variables must only include variables that are not INFLUENCED by the treatment, • i.e. variables that have been collected BEFORE the intervention • >> If not: Problem of reverse causality!**Control Variables (3)**• What matters in terms of power is ,the residual variation after controlling for those variables • so just replicate the steps described above within strata • It may help stratifying along dimension that we know from previous studies are important for the effects of the program • Example: we might expect to have differential effects by gender or age groups >> This may help understand “non-response” rates**Wrap up**• Power calculations look scary but they are just a formalization of common sense • At times we do not have the right information to conduct it very properly • However, it is important to spend effort on them: • Avoid launching studies that will have no power at all: waste of time and money, potentially harmful • Devote the appropriate resources to the studies that you decide to conduct (and not too much)**Relation with Confidence Intervals**• A 95% confidence interval for an effect size tells us that, for 95% of any samples that we could have drawn from the same population, the estimated effect would have fallen into this interval • If zero does not belong to the 95% confidence interval of the effect size we measured, then we can be at least 95% sure that the effect size is not zero • The rule of thumb is that if the effect size is more than twice the standard error, you can conclude with more than 95% certainty that the program had an effect**Standardized Effect Sizes (but this is a 2OP here)**• Sometimes impacts are measured as a standardized mean difference • Example: outcomes in different metrics must be combined or compared • The standardized mean effect size equals the difference in mean outcomes for the treatment group and control group, divided by the standard deviation of outcomes across subjects within experimental groups

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