C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Between Group & Within Subjects Designs Mann-Whitney Test Wilcoxon Test Matching Samples Using Ranks

C82MCP Diploma Statistics School of Psychology University of Nottingham 2 Between Group & Within Subjects Designs. There are two forms of manipulation of the independent variable. All the conditions can be applied to the same subject All the conditions applied to different groups of subjects. In a situation where different groups of subjects do different things then we have a between group design. In a situation where the same group of subjects do different things then we have a within subjects design

C82MCP Diploma Statistics School of Psychology University of Nottingham 3 Two Level Non-Parametric Tests Whenever we vary an independent variable, we can have many different levels of that variable. There are two commonly used two level non-parametric tests. The Mann-Whitney Independent Samples Test. The Wilcoxon Related Samples Test. The Mann-Whitney is used when we have a between groups (independent samples) design The Wilcoxon is used when we have a within subjects (or matched/related samples) design.

C82MCP Diploma Statistics School of Psychology University of Nottingham 4 The Mann-Whitney Independent Samples Test. The Mann-Whitney Test is a non-parametric test The Mann-Whitney Test is designed to look at differences between treatments in a between groups design The Mann-Whitney Test is used when there are only two treatment levels.

C82MCP Diploma Statistics School of Psychology University of Nottingham 5 The Rationale of the Mann-Whitney Test. If we arrange all the data into an ascending sequence: When the null hypothesis is true we would expect the scores of the two groups to be randomly distributed When the null hypothesis is false we would expect the scores of the two groups systematically distributed in the sequence.

C82MCP Diploma Statistics School of Psychology University of Nottingham 6 Example Data Imagine an experiment that compared two groups of subjects on learning lists of words and pictures.

C82MCP Diploma Statistics School of Psychology University of Nottingham 7 Example Data We can imagine all these scores placed in a straight line and given a rank with the lowest score ranked 1 and the highest score ranked 12.

C82MCP Diploma Statistics School of Psychology University of Nottingham 8 Example Data We can put these ranks into a table. If the null hypothesis is true the rank totals should be the same. If the null hypothesis is false the rank totals should be different

C82MCP Diploma Statistics School of Psychology University of Nottingham 9 The Mann-Whitney U We calculate a statistic, known as the Mann-Whitney U which provides an estimation of the deviation of the sums of the ranks from what could be expected by chance. where

C82MCP Diploma Statistics School of Psychology University of Nottingham 10 Evaluating the Null Hypothesis As with other tests, we compare Uobserved against Ucritical The critical value of U is found in the statistical tables The table has n 1 and n 2 along the top row and the first column respectively. Find the critical value of U by moving along to the column that has the number of subjects in one group and down that column to the number of subjects in the other group. For the Mann-Whitney test the null hypothesis is rejected when U observed  U critical

C82MCP Diploma Statistics School of Psychology University of Nottingham 11 Evaluating the Null Hypothesis For the example data The critical value of U is 5 Since 3<5 there is significant difference between the groups. We reject the null hypothesis and conclude that subjects can recall more pictures than words

C82MCP Diploma Statistics School of Psychology University of Nottingham 12 The Wilcoxon Related Samples Test. The Wilcoxon Test is a non-parametric test The Wilcoxon Test is designed to look at differences between treatments in a within subjects design The Wilcoxon Test Test is used when there are only two treatment levels.

C82MCP Diploma Statistics School of Psychology University of Nottingham 13 The Rationale of the Wilcoxon Test. After collecting data an experimenter can usually tell two things in a two level within subjects design: Which of the two treatment scores is "greater than" the other. The absolute size of the difference between the treatment scores. The Wilcoxon test takes advantage of this knowledge.

C82MCP Diploma Statistics School of Psychology University of Nottingham 14 Example Data Imagine that we had conducted an experiment that looked at the effect of interruptions on the ability to solve seriation problems. We could provide subjects with a set of seriation tasks i.e. If bill is taller than fred and fred is taller than jim is bill taller than jim? On some of the tasks the subject is allowed to complete the task and on other tasks the experimenter interrupts the subject before letting them continue.

C82MCP Diploma Statistics School of Psychology University of Nottingham 15 Example Data We could then count the number of correct answers that the experimental subjects made.

C82MCP Diploma Statistics School of Psychology University of Nottingham 16 Example Data The Wilcoxon uses two pieces of information: is one score greater than another? what is the size of this difference?

C82MCP Diploma Statistics School of Psychology University of Nottingham 17 The Rationale of the Wilcoxon Test If the null hypothesis were true we would expect some of the differences to be due to the not interrupted condition and some of them due to the interrupted condition. The sign of the differences would be about equal. There would be an equal number of positive and negative differences If the null hypothesis were true we would also expect the size of the differences to be about equal for the interrupted and not interrupted conditions. The absolute total of positive differences would be equal to the absolute total of negative differences

C82MCP Diploma Statistics School of Psychology University of Nottingham 18 The Rationale of the Wilcoxon Test If we rank the absolute values of the scores we expect the sum of the ranks for the positive differences to be about the same as the sum of the ranks for the negative differences. That is, we rank the scores ignoring their sign. So to use this test we have to define two statistics. T+ = the sum of the ranks of the positive differences T- = the sum of the ranks of the negative differences

C82MCP Diploma Statistics School of Psychology University of Nottingham 19 Example Data For the example data we get: T + = = 32 T - = 1+3 = 4

C82MCP Diploma Statistics School of Psychology University of Nottingham 20 Evaluating the Null Hypothesis The null hypothesis states that (T+) and (T-) should be about equal. We can use a table of critical values to estimate the probability that the results occur by chance. First we identify the smaller value T + < T - T - < T + We call the smaller value W. The null hypothesis is rejected when W observed  W critical

C82MCP Diploma Statistics School of Psychology University of Nottingham 21 Evaluating the Null Hypothesis For the example data T - = 4; T + = 32 T - < T + Therefore W=4 The critical value of W is 3 Since W observed is not less than W critical we fail to reject the null hypothesis. There is no significant difference between the situations There is no evidence in this experiment to show that subjects perform worse on seriation tasks when interrupted

C82MCP Diploma Statistics School of Psychology University of Nottingham 22 Matching Samples Sometimes observed differences between two groups of subjects in a between groups design are not due to the independent variable. For example, we might look at two groups of subjects who have been taught in different ways. We might find that there is a difference between the groups. This could be due to the different teaching strategies or it could be due to the two groups being different to start off with. One way to overcome this difficulty kind of difficulty is to use two related samples.

C82MCP Diploma Statistics School of Psychology University of Nottingham 23 Matching Samples A matched sample is when a group of subjects have been matched on relevant criteria, e.g. age, race, gender, IQ, etc. This groups of subjects are then treated as a single subject for the purposes of analysis One subject from each group completes one level of the independent variable.

C82MCP Diploma Statistics School of Psychology University of Nottingham 24 Using Ranks Whenever you have tied ranks then take the average of the range of ranks that the ties cover and allocate this to the value of the ties. Do not change the other ranks.

C82MCP Diploma Statistics School of Psychology University of Nottingham 25 Using Ranks Whenever you are looking at ranked differences in a within subjects (matched) design if there is no difference then remove the subject (matched group) from the analysis.

C82MCP Diploma Statistics School of Psychology University of Nottingham 26 Summary. The Mann-Whitney test is a non-parametric test used for between group two-group experiments. The Wilcoxon test is a non-parametric test used for two sample within subject (matched) experiments.