How to Perform Column Comparisons with Missing Repeated Measures Data

This article describes alternative approaches to statistical testing with missing data when repeated measures are tested against one another.

Requirements

• A data file with repeated measures data with missing data issues in the repeated measures.

Repeated measures data is data where a respondent has provided two or more evaluations and there is a need to compare them (e.g., ratings of satisfaction at different points in time, ratings of the appeal of two different products).

Typically, such data appears in Q in a Pick One - Multi question, a Pick Any - Grid, or a Number - Grid.

Examples

Example 1: Lots of missing data

In this example, there is lots of missing data, and no respondents have seen both A and B. Consequently, as Q defaults to only performing testing with respondents that have complete data, no tests are performed (as indicated by the -). In this example, conducting an independent samples test is likely the correct approach (see the instructions below).

Example 2: Some missing data

The table below shows that a higher proportion of respondents said they would buy Product B (20%) than Product A (16%). But, the column comparisons tell the opposite story, with Product A shown to have a significantly higher Buy score than Product B. While this result seems to be a paradox, it is not an error and is instead a consequence of a serious missing data problem.

The Column n at the bottom of the table shows that the sample size in Product A is 142, compared to 191 for B. When the data is filtered to include only respondents that have complete data, we get the table below. This is the table that Q has used in the "background" when performing the test of columns A and B. Note that with this table we can see that the first column's score is higher than that of the second column (16% versus 12%).

Understanding the logic of Q

When faced with missing repeated measures data, some people prefer to conduct testing using the numbers shown in the tables. That is, in the case of Examples 1 and 2, Q should just compare the numbers shown. There are a number of reasons why Q does not do this and instead filters the data:

• Consistency
• Filtering the data is the orthodox solution in statistical testing
• It increases the user's chance of detecting a problem

Consistency

When there is no missing data, Q performs the standard repeated measures tests. Consequently, it would be confusing if Q did something different when some missing data exists.

Filtering the data is the orthodox solution in statistical testing

With repeated measures data such as this, the orthodox statistical treatment of the data is to perform testing only using respondents that have no missing data. Examining the data from Example 2 gives some insight into why this is the orthodox approach.

Looking at the data for Product B in Example 2, 20% of 191 Buy, whereas in the second table 12% of the 140 Buy. From this, we can deduce that the 51 respondents with missing data were, on average, much more likely to have said they would buy. The following table compares the Buy and Not buy data for product B according to whether or not there is any missing data, and shows that 41% of those with some missing data said Buy.

Thus, the 20% shown on the original table is a weighted average of two very different groups' data: the group with missing data who have given much higher Buy ratings, and the people who evaluated both products, who gave lower ratings for Product B. By default, Q ignores the data from those with missing data. There are a number of methodological justifications for this:

• The data from respondents that have seen both products may be more reliable than data from respondents that have only seen one. This is, of course, only a conjecture.
• There are no widely-recognized statistical tests that take into account the missing data. This is discussed in more detail below.
• Filtering the data and employing a repeated measures test provides more statistical power than using an independent samples tests (independent samples tests are discussed below).
• Where there is missing data, a valid analysis requires assumptions to be made about the causes of the missing data. Some possible assumptions and their relevant implications are:
  • Data that is missing can be considered to be missing completely at random. If this assumption is correct, then it is safe to ignore the respondents with missing data (although statistical power may be reduced), as is done by default by Q.
  • Respondents with missing data are intrinsically different (i.e., the missing data is not ignorable). In surveys, this is often the case (e.g., people with missing data may be people that have missing data because they are less experienced in a category). If this is the case, a valid analysis would need to involve estimating how the 51 respondents with some missing data would have evaluated each of Product A and Product B. That is, if the missing data is not ignorable, it means that it is invalid to compare the original numbers in the first table (20% versus 16%) because the 16% is a biased estimate due to not taking into account the missing data.

It increases the user's chance of detecting a problem

A criticism of the way that Q performs the testing is that "effectively the numbers show one thing but the significance testing shows another". This is an accurate description of how Q works, but the testing with Q was deliberately written to achieve this outcome. The initial table in Example 2 shows the results of 20% and 16% because these are the numbers in the data. However, the test that is shown is Q's attempt to produce the best test possible given the data. Were Q to instead provide a test consistent with the data, it would mean that the test would, more often than not, be invalid. Further, there would be no way for anybody reading the table to identify the problem, as the numbers would appear to make sense. By contrast, by providing the paradoxical result, the user is able to identify that something is wrong and investigate the problem further.

A second cue to the existence of the missing data problem is that Q shows a range of the missing data at the bottom-right of the screen (i.e., base n = from 142 to 191).

Alternative approaches to statistical testing with missing data

Next

How to Set Statistical Assumptions When Setting Up Projects

How to Show Statistical Significance in Q