Not all sample size issues are directly related to power. Accuracy in parameter estimation (AIPE) is also a function of sample size, that is, the larger the sample size the smaller the confidence interval for a parameter estimate. Accuracy in parameter estimation allows you to specify the size of confidence interval you want to achieve and, in return, gives you the sample size needed to achieve that confidence interval. As Kelly and Maxwell (2003) state, "The AIPE approach yields precise estimates of population parameters by providing necessary sample sizes in order for the likely widths of confidence intervals to be sufficiently narrow."
In this unit we will illustrate how to do an AIPE analysis for a multiple regression model that has two control variables, one categorical research variable and one continuous research variable, with the focus being on the confidence interval for the continuous research variable.
We will be using the same data analysis example that was used in the unit on multiple regression power analysis. In that analysis, a school district is designing a multiple regression study looking at the effect of gender, family income, mother's education and language spoken in the home (3 levels, 2 dummy variables) on the English language proficiency scores of Latino high school students. Mother's education is the primary research variable that measures the number of years that the mother attended school. It is a continuous variable ranging from 4 to 18 years.
When we ran the power analysis for testing the parameter for mother's education, we came up with sample sizes of 108, 138 and 182 for power values of .7, .8 and .9 respectively. We can check these values against the sample size needed to achieve a researcher specified confidence interval.
To conduct an AIPE analysis we will use the Stata program aipe (findit aipe) (see How can I use the findit command to search for programs and get additional help? for more information about using findit). This program was written by UCLA Academic Technology Services as an implementation of the approach outlined in the Kelley and Maxwell (2003) article.
In this analysis we want to determine how large a sample will be needed to have a confidence interval on the regression coefficient that extends 0.1 above and below the point estimate. A confidence interval can be calculated as
For this analysis, we believe, based on previous research, that the R2 for the full model will be about 0.48 and that the R2 for mother's education with the other predictors will be about 0.4. The total number of predictors in the model is 5 and we want a confidence interval half-width of 0.1 with an alpha level of 0.05.
aipe, r2(.48) r2xx(.4) w(.1) p(5) alpha(.05) Accuracy in Parameter Estimation p = 5 -- number of predictor variables in full model alpha = .05 -- alpha level for confidence interval w = .1 -- confidence interval half-width R2 = .48 -- R-squared for full model R2xx = .4 -- R-squared for target predictor with other predictors quan = .8 -- quantile of chi-square distribution AIPE sample size N = 339 -- n needed for CI half-width of w = .1, 50% of time Nm = 366 -- n needed for CI half-width of w = .1, 100*(quan)% = 100*(.8)% = 80% of time
A confidence interval with a half-width of 0.1 will require either 339 or 366 students, depending on the percent of time that the half-width is likely to occur. These sample sizes are considerably larger than those from the power analysis for mother's education.
Let's see what making the confidence interval wider does for the sample size. We will rerun the analysis using a half-width of 0.15.
aipe, r2(.48) r2xx(.4) w(.15) p(5) alpha(.05) Accuracy in Parameter Estimation p = 5 -- number of predictor variables in full model alpha = .05 -- alpha level for confidence interval w = .15 -- confidence interval half-width R2 = .48 -- R-squared for full model R2xx = .4 -- R-squared for target predictor with other predictors quan = .8 -- quantile of chi-square distribution AIPE sample size N = 154 -- n needed for CI half-width of w = .15, 50% of time Nm = 174 -- n needed for CI half-width of w = .15, 100*(quan)% = 100*(.8)% = 80% of time
The required sample sizes are much smaller for a half-width of 0.15 than for 0.1.
Based on the accuracy in parameter estimation analyses and taking into consideration that the researchers want a very narrow confidence interval half-width of 0.1, we will go with a planned sample size of 340 students. A larger sample size would make the specified confidence interval more likely, but having a half-width of 0.1 50% of the time was considered sufficient by the researchers involved in the study.
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