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Estimating the Effects of Treatment on Outcomes with Confidence Sebastian Galiani Washington University in St. Louis

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Parameters of Interest Two parameters of interest widely used in the literature: Average Treatment Effect Average Treatment Effect on the Treated Under randomization and full compliance, they coincide.

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Randomization In the absence of difficulties such as noncompliance or loss to follow up, assumptions play a minor role in randomized experiments, and no role at all in randomized tests of the hypothesis of no treatment effect. In contrast, inference in a nonrandomized experiment requires assumptions that are not at all innocuous.

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Quasi-Experimental Designs If randomization is not feasible, we need to rely on quasi-experimental methods. In our case, the most promising strategy would be a Generalized Difference in Differences strategy.

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Parameters of Interest We might want to response the following questions: What is the effect of the intervention on a given outcome on a given population? What is the effect of the intervention on a given outcome on those that self-select as users of the facilities? The power for identifying the first parameters might be lower than the power for the second parameter –identified by IV Methods.

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Distance to Facilities and Sampling Think about stratifying the sample by distance to facilities, and over-sample households residing near facilities.

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Testing absence of Treatment Effects. Type I Error Once we have chosen a Type I error rate: , the null hypothesis ( T - C = 0) is rejected whenever the statistics of contrast |t| > t* ; where t* is the (critical) value of t that defines the /2 percentile of the distribution of t. 0 -t* /2 t* /2 No rejection Reject Null

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Type II Error Now consider an alternative hypothesis: T - C = d Under this alternative hypothesis, the t-statistic will have a different distribution. If the alternative hypothesis is true, we want to reject the null hypothesis as often as possible. To fail to do so would be a Type II error. We want to restrict the probability of this type of error to Then will be the type II error rate of the test. And 1- will be the power of the statistical test. The power of a statistical hypothesis test measures the test's ability to reject the null hypothesis when it is actually false.

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Type II Error 0-t* /2 t* /2 No rejection Reject Null Distribution of t under the null Distribution of t under the alternative Power 1- There is a trade-off between Type I and Type II errors

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Type I and II Errors 0-t* /2 t* /2 No rejection Reject Null Distribution of t under the null Distribution of t under the alternative Power 1- Both errors can be simultaneously reduced if the dispersion of the statistics is reduced.

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Simple Clinical Trial In this design m members are allocated to each condition: treatment and control. The observed value to the i-th member in the l-th condition is a function of the grand mean and the effect of the l-th condition; any difference between the observed and the predicted value is allocated to the residual error. The intervention effect is C l. Its estimate is:

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Simple Clinical Trial Under the null hypothesis, H 0 : C l = 0, Let estimate the variance of. Assuming that the residual error is distributed Gaussian, the intervention effect is evaluated using a t-statistic with the appropriate df. The researcher determines the desired Type I and II error rates (say 5% and 20%, respectively). The researcher expects a negative intervention effect but would be concerned about a positive effect; as a result, she chooses a two-tailed test.

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Simple Clinical Trial Given the random assignment of members to treatment and control conditions, it is reasonable to assume that the two study conditions are independent. Then: The estimated variance of a single condition mean is: If we assign the same number of members in each condition, the variances in the two conditions are assumed to be equal.

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Simple Clinical Trial Then, the t-statistic is estimated as: The parameters appearing in this formula are relatively easy to estimate using data from previous reports, from analyses of existing data or from preliminary studies.

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Simple Clinical Trial True type I and II errors rates will be and respectively if: 0-t* /2 t* /2 No rejection Reject Null Distribution of t under the null Distribution of t under the alternative Power 1- -t* /2 -t* H0H0 HAHA

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Or: This expression is general to any design. We need: –Desired type I and II error rates. –The expected magnitude of the treatment effect. –The expression for the variance of the estimated treatment effect, which is a function of the sample size. We can express any of these variables in terms of the others. General Expression

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Simple Clinical Trial Sample size:

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Sample Size in GRT Assume that there is only one individual per household. Probability that an individual has diarrhea: Individual i, in group k, assigned to condition l. Within each condition: the variance of any given observation is: Where stands for the variance within groups and for the variance between groups

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Sample Size in GRT Consider first the group mean. If that mean were based on m independent observations, the variance of that mean would be estimated as: However, because the members within an identifiable group almost always show positive intra-class correlation, those observations are not independent. In fact, only the variance attributable to the individual effect will vanish as m increases. The variance attributable to the group effect will remain unaffected.

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Sample Size Then, the variance of the group mean is: Where, m stands for the number of households per group and ICC for the intra-group correlation. The variance of the condition mean is: Where g is the number of groups in each condition

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Sample Size When ICC>0, the variance of the condition mean is always larger in a GRT than in a study based on random assignment of the same number of individuals to the study conditions. Statistic of interest: Variance of the statistic:

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Sample Size Given a moderate number of groups per condition, the t-statistic to asses the difference between condition means is: It is distributed t-student with g T +g C -2 degrees of freedom

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Sample Size Sample size: –Number of groups per condition: –Number of household per group (it requires a couple of iterations):

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Sample Size When each household has more than one observation, we need to perform the following correction: Where a is the number of observations per household and is the intra-household correlation. See extreme cases or

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Pretest - Posttest: Repeat Observations on Groups Data are collected in each condition before and after the intervention has been delivered in the intervention condition. There are repeated observations on the same groups

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Pretest - Posttest: Repeat Observations on Groups The model: The observed value for the i-th member nested within the k-th group and l-th condition and observed at the j-th time is expressed as a function of the grand mean, the effect of the l-th condition, the effect of the j-th time, the joint effect of condition and time, the realized value of the k- th group, the joint effect of group and time. Differences between this predicted value and the observed value are allocated to the residual error

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Pretest- Posttest: Repeat Observations on Groups Treatment effect: Dif-in-dif There are two sources of variation among the groups: –Variation due to group effect –Variation due to the interaction group x time The first difference eliminates the first source of variation. The group mean is: This model can be easily transformed in the basic GRT

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Pretest - Posttest: Repeat Observations on Groups The variance of the group mean is: Following the same steps as before… The variance of the intervention effect can be written as: Sample size can be solved as before

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Pretest - Posttest: Repeat Observations on Members Data are collected in each condition before and after the intervention has been delivered in the intervention condition. There are repeated observations on the same members

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Pretest- Posttest: Repeat Observations on Members The model: The observed value for the i-th member nested within the k-th group and l-th condition and observed at the j-th time is expressed as a function of the grand mean, the effect of the l-th condition, the effect of the j-th time, the joint effect of condition and time, the realized value of the k-th group, the realized value of the i-th member, the joint effect of group and time and the joint effect of member and time. Differences between this predicted value and the observed value are allocated to the residual error

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Pretest- Posttest: Repeat Observations on Members Treatment effect: Dif-in-dif There are three sources of variation among the members: –Variation due to member effect –Variation due to the interaction member x time –Error term The first difference eliminates the first source of variation.

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Pretest- Posttest: Repeat Observations on Members Taking differences by members: This model can be easily transformed in the basic GRT The variance of the intervention effect can be written as: Sample size can be solved as before

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