Assignment II for Biostats Course VHM 801 at AVC - Winter semester 2018

The assignment is worth 15% of the final course mark. Please be aware that by handing in the home assignment you implicitly acknowledge to have read and accepted the instructions for home assignments as described on the VHM 801 homepage.

The assignment uses information from studies conducted in the early 1990s about the impact of chess training on reading skills. The researchers hypothesized that chess play would develop skills of importance for reading, and hence also lead to improvements in reading ability. Students who had taken a standardized reading (DRP) test and had shown interest in chess (according to certain criteria) were offered chess instruction after school; participation was voluntary. After the chess instruction period, the students took the DRP test again; the testing was standardized to a one year period between the pre- and post-tests. The scale of DRP tests is adjusted to the grade level (so that the 50% percentile corresponds to the grade median for all students), and the scores obtained in the study were converted to grade-level percentiles. Therefore, a test score of e.g. 28 means that the student's result corresponded to the 28% percentile for her/his grade. A total of 53 students participated in the part of the study presented here.

A dataset for the study is available in Minitab format and as a comma-separated file, for import into Stata and other statistical software. The data contain the variables: subject id, pre-score (percentile) of the DRP test, and post-score (percentile) of the DRP test. The home assignment has six questions (a)-(f) which should all be answered.

1. A question of interest was whether the participants, prior to chess instruction, were representative for their grade(s) in terms of reading skills. Use statistical inference to assess whether participants' reading skills were systematically lower or higher than 50%. State your results and conclusions carefully. Moreover, list and discuss critically the assumptions the inference is based on; here, as well as in following questions, include as much information from a descriptive data analysis as necessary for your arguments.

2. The main focus of interest was the comparison of reading skills prior to and after chess instruction. Should the test scores before and after chess instruction be considered as paired or independent samples? Explain your reasoning. Additionally describe, as well as you can from the information provided, the population we can make inference about based on the data and the results (to be obtained).

3. Estimate the mean improvement (possibly negative) in reading score percentiles after the chess instruction. Calculate 90% and 95% confidence intervals for the mean improvement, and interpret these intervals carefully. Make sure to explain clearly how the two intervals differ in their interpretation.

4. Use a statistical test to assess whether there was an improvement in the DRP scores after chess instruction. State your hypotheses, report the value of your test statistic, and give the P-value for the test. Provide your conclusions, both statistical and subject matter. Explain also here which assumptions the inference for this and the previous question (c) rely on, and discuss briefly their validity.

5. According to one source discussing these data, the data for subject no. 7 could be considered as a "mild outlier". Inspect the data and try to deduce how this conclusion may have been reached - do you agree with the assessment? (Hint: No statistical analysis is required for the discussion.) Here we will tentatively also perform an analysis without the data for subject for no. 7, in order to explore how they affect the results. Specifically, repeat the analyses of parts (c) (95% CI only) and (d) for the reduced dataset. Did removing the (possible) outlier change your results? Comment on your findings.

6. One way to quantify whether the improvement of 42 percentage points for subject no. 7 should be considered as an outlier is to compute the probability that such an extreme observation happened by chance only. This question will take you through the steps to approximately compute that probability. The first step is to set up a statistical model on which the calculation will be based. Here we will assume a normal distribution with mean and standard deviation equal to the corresponding sample values in the sample of improvements (post-score minus pre-score) where the value 42 has been removed. Determine these estimates. Next compute the probability that one observation from this (normal) distribution differs as much from the mean as the value 42 does (either on the positive or negative side). Finally, compute the probability that in a sample of 53 independent observations from this distribution at least one of them differs as much from the mean as the value 42 does. Interpret the computed probability - does it seem reasonable that the value 42 happened by chance only? As an optional bonus question (which may offset loss of points in other questions), discuss critically the assumptions involved in the calculation and whether they might have affected (biased) the results, and if so, indicate also the direction of the bias.
   (Note: Other methods for statistical assessment of outliers exist, but it is not expected or recommended that you include those in your discussion. If you nevertheless decide to include methods not covered by Sessions 1-7 of VHM 801, you have to explain them in enough detail to demonstrate that you understand how they work and which assumptions they are based on.)