21.4.3: Effects of Rounding

Problem:

Determine the effect of rounding on the appropriateness of the one-sample t-test.

Rounding can significantly affect statistical decisions because it alters the precision of numerical data (the inputs), which can lead to changes in calculated values such as means, variances, or p-values (the outputs). In hypothesis testing, for instance, rounding may cause a test statistic to fall on the border of a critical value, potentially shifting the conclusion about rejecting or failing to reject the null hypothesis. Similarly, in regression analysis, rounding predictor or response variables can affect the accuracy (and precision) of parameter estimates and the overall model fit.

These impacts are particularly pronounced in datasets with small sample sizes or values that are close to decision thresholds. Therefore, careful consideration of rounding practices is essential to ensure that statistical conclusions remain valid and reliable. This example explores the effects of rounding on the one-sample t-test.

Solution:

This leaves a lot of decisions to us. The one-sample t-test requires data being generated from Normal distribution. So, let's generate data from a Normal(5, 1) distribution. We need to round the data, so we will need to use the round function. Finally, we will need to determine if the distribution of the resulting p-values is standard Uniform distribution.

The following code does this. Make sure you understand every piece of this code and why the output tests the effects of rounding.

pval = numeric()

for(i in 1:1e4) {
 x = rnorm(10, m=5, s=1)
 y = round(x)
 pval[i] = t.test(y, mu=5)$p.value
}

ks.test(pval,"punif")
binom.test(sum(pval<0.05), n=length(pval), p=0.05)

　

The Kolmogorov-Smirnov test indicates that the distribution of the p-values is not standard Uniform. Thus, rounding in this situation breaks the t-test. Note that if we only care about α = 0.05, we would use the Binomial test results. Given that p-value, I would still conclude that I should not use the t-test in this situation.

What about increasing the sample size from 10 to 50? The Kolmogorov-Smirnov test still indicates that the test is no longer acceptable. Note that if all we care about is α = 0.05, then the t-test does appear to be appropriate under these conditions.

What about increasing the variability in the data? Having a wider spread to the data may make the rounding less important. Let's change the standard deviation from 1 to 10 (and return the sample size to 10). From my run, the distribution of the p-value is still not standard Uniform, but the rejection rate for α = 0.05 is close enough to 0.05.

Extension:

What if we increase the sample size back to 50? Show that the distribution of the p-values is close enough to standard Uniform that the rounding does not affect the quality of the t-test conclusions.