![]() “ There is no change in ability scores after training”). The exact wording of the written-out version should be changed to match whatever research question we are addressing (e.g. ![]() As with our other null hypotheses, we express the null hypothesis for paired samples t-tests in both words and mathematical notation. Let’s be clear, H0: μD = 0 does not say that everyone in the population will stay the same it only says that on average, the entire population will show a mean difference of 0. In this text, we will refer to paired samples, though the appearance of any of the other names throughout this chapter should not be taken to refer to a different analysis: they are all the same thing. As such, all of these names are equally appropriate, and the choice of which one to use comes down to preference. ![]() What all of these names have in common is that they describe the analysis of two scores that are related in a systematic way within people or within pairs, which is what each of the datasets usable in this analysis have in common. It is important to point out that this form of t-test has been called many different things by many different people over the years: “matched pairs”, “paired samples”, “repeated measures”, “dependent measures”, “dependent samples”, and many others. Such datasets and analyses are the subject of the following chapter. This type of analysis would not work if we had two separate samples of people that weren’t related at the individual level, such as samples of people from different states that we gathered independently. This is what makes the analysis of change unique – our ability to link these measurements in a meaningful way. That is, a single observation or data point is comprised of two measurements that are put together into one difference score. In both of these types of data, what we have are multiple scores on a single variable. To calculate improvement or any other difference score, we must measure only a single variable. An important factor in this is that the participants received the same assessment at both time points. There’s also one improvement score of 0, meaning that the training did not help this employee. Notice that the lowest scoring employee before the training (with a score of 1) improved just as much as the highest scoring employee before the training (with a score of 8), regardless of how far apart they were to begin with. What we are not interested in is how good they were before they took the training or after the training. We want to see positive scores, which indicate that the employees’ performance went up. This third column is what we look at when assessing whether or not our training was effective. the score after minus the score before) represents improvement in the employees’ ability. The difference between these scores (i.e. Table 1 shows scores on a quiz that five employees received before they took a training course and after they took the course. Raw and difference scores before and after training. If the average difference between scores in our sample is very large, compared to the difference between scores we would expect if the member was selected from the same population then we will conclude that the individuals were selected from different populations. The absolute value of our measurements does not matter – all that matters is the change. When we analyze data for a repeated research design, we calculate the difference between members of each pair of scores and then take the average of those differences. This means that all subjects participate in each treatment condition. Data consist of two scores for each individual. This is a repeated sample research design, where a single group of individuals is obtained and each individual is measured in two treatment conditions that are then compared. In each of these cases, we measure a single variable at different times, and what we are looking for is whether or not we get the same score at time 2 as we did at time 1. Sometimes we want to see if change occurs naturally, and other times we are hoping for change in response to some manipulation. Researchers are often interested in change over time. This is a very powerful thing to do, and, as we will see shortly, it involves only a very slight addition to our existing process and does not change the mechanics of hypothesis testing or formulas at all! Change and Differences Specifically, we will look at how the value of a variable, within people, changes across two timepoints. Now, we will look at a slightly different type of data that has new information we couldn’t get at before: change. So far, we have dealt with data measured on a single variable at a single point in time, allowing us to gain an understanding of the logic and process behind statistics and hypothesis testing.
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