7.4.1 The Sign Test

Tests of Difference: Sign Test

A sign test is used when looking at paired or related data. The two related pieces of data should come from repeated measures design or matched pairs design. The data should be nominal.

Step 1: State the Hypotheses

• Alternate Hypothesis: Blood pressure (measured after the stress management programme) will be lower after participants undergo the stress management programme.
• Null Hypothesis: There will be no difference in blood pressure before and after the stress management programme. Any difference found will be due to chance.

Step 2: Record the data. Step 3: As the sign test uses nominal data, we need to convert our data by working out which participants had a lower BP after the stress management programme, and which had a higher BP. For each PP subtract the After BP from the before BP and record the sign, whether it was positive or negative. Step 4: Count the number of positive signs and the number of negative signs. The calculated S value will be the lower of these two numbers. (No' of – signs …… and No' of + signs….)

Participants	Before BP	After BP	Difference	Sign
1	05	100
2	110	99
3	118	120
4	101	95
5	90	89
6	95	101
7	125	125
8	87	85
9	96	108
10	92	100
11	88	80
12	120	124

Observed value of S: _____ (is the smaller sign value: add up the +s and the – s & choose smallest)

Step 5:

Find the critical value of S so we can compare it to the observed value of S.

1. N= the total number of scores (ignoring any 0 or = values)
2. One-tailed or two-tailed hypothesis
3. p= 0.05 usual level of probability chosen (1 in 20 likelihood results are due to chance, 5%) N=_______

One-tailed or two-tailed

Critical value of S: _____

Step 6: For the data to be significant (and the alternative hypothesis to be accepted) the calculated value of S must be EQUAL to or LESS THAN the critical value identified.

The results are / are not significant as the observed value (calculated value) of S = ……………. is higher / lower than the critical value …………., where N=………………… for a ……………. -tailed hypothesis, with a p ≤ …………. level of significance, the null hypothesis can be rejected / accepted, and the alternative hypothesis rejected / accepted.

Tests of Difference: Wilcoxon T Test

A test for a significant difference between two sets of scores. Data should be ordinal using a related design (repeated measures or matched pairs).

Step 1: State the hypotheses

Step 2: work out difference between condition A and condition B. State whether the difference is a positive or negative number.

Step 3: rank the differences – ignore the participants who gain a difference score of 0 and ignore +/- signs. Rank the lowest difference as 1 and work from lowest to highest. If there are any of the same numbers, they will take up two ranks but will be ranked the same (for example if you have two participants who have a score 1 of in the difference column, they take up ranks 1 and 2 but are both ranked 1.5 and the next number will take up rank 3 and so on).

Step 4: Find the T value (usually provided in the stem) and compare it to the critical value. When identifying the critical value, N= number of participants – number of participants with 0 differences. For the Wilcoxon T-Test, the observed value must be equal to or less than (p<0.05) the critical value to gain significance.

Step 5: Write a statement of significance.

The results are / are not significant as the observed value (calculated value) of T = ……………. is higher / lower than the critical value …………., where N=………………… for a ……………. -tailed hypothesis, with a p ≤ …………. level of significance. The null hypothesis can be rejected / accepted, and the alternative hypothesis rejected / accepted therefore [relate back to hypothesis].

Test of difference: Mann-Whitney U Test

A test for a significant difference between two sets of scores. Data should be ordinal using an unrelated design (independent groups).

Step 1: State the hypotheses

Step 2: Rank the ratings, considering both groups at the same time (ignore the fact that they are in different conditions). The lowest number has a rank of 1. In the case where two data items are the same, add up the rank and give the mean for those ranks (for example, if the rating of 12 appears four times in table at rank position 7, 8, 9, and 10, they are all given the rank 8.5 à (7 + 8 + 9 + 10) ÷ 4 = 8.5). Calculate the sum of ranks.

Step 3: find the $n_1, n_2$, and U value and compare this to the critical value. For the Mann-Whitney test, the value of U must be equal to or less than the critical value for significance to be shown.

Step 4: write a statement of significance.

The results are / are not significant as the observed value (calculated value) of U = ……………. is higher / lower than the critical value …………., where $N_1$=…………… and $N_2$= ………… for a ……………. -tailed hypothesis, with a p ≤ …………. level of significance. The null hypothesis can be rejected / accepted, and the alternative hypothesis rejected / accepted therefore [relate back to hypothesis].

Parametric Tests

Parametric tests are powerful statistical tests meaning that they are better able to detect a significant effect. This is because they are calculated using the actual scores rather than the ranked scores. However, this sensitivity can also be a problem if the data is inconsistent or erratic.

Parametric tests are also robust, meaning they can cope with data which do not fully meet the three criteria. The only essential criterion is that the data must be interval level. If this criterion is not met, then a non-parametric test needs to be used instead. Non-parametric tests are calculated using ranks, which means they are less sensitive but better able to cope with any inconsistency in the data. The level of measurement is interval, the sample must be assumed to be normally distributed and there must be homogeneity of variance (identical or similar standard deviations).

Parametric Tests of Difference: Unrelated t-test

A parametric test for the difference between two sets of scores. Data must be an interval with independent groups design.

Step 1: state the hypotheses

Step 2: place raw data in the table

Step 3: find the calculated value of t and compare it to the critical value ($df$ value = ($N_A$ + $N_B$) – 2)

For the Unrelated T-test the observed value must be EQUAL to or GREATER than (p ≤ 0.05) the critical value to gain significance.

Parametric Tests of difference: Related t-test

Parametric test of difference between two sets of scores. Data must be an interval with related design – repeated measures or matched pairs.

Step 1: state the hypotheses

Step 2: place raw data in the table

Step 3: find the calculated value of t and compare it to the critical value ($df$value = N – 1)

For the Related T-test, the observed value must be EQUAL to or GREATER than (p ≤ 0.05) the critical value to gain significance.

Write a Statement of Significance: The results are / are not significant as the observed value of T = ……………. is higher / lower than the critical value …………., where N=………………… for a ……………. -tailed hypothesis, with a p ≤ …………. level of significance, the null hypothesis can be rejected / accepted, and the alternative hypothesis rejected / accepted.

Parametric Tests of Correlation: Pearson's R test

Parametric correlation test when data is at interval level.

Step 1: state the hypotheses

Step 2: place raw data in the table

Step 3: find the calculated value of r and critical value ($df$ value = ($N_A + N_B$) – 2)

For Pearson's r, the observed value must be EQUAL to or GREATER than (p ≤ 0.05) the critical value to gain significance.

Write a statement of significance: The results are / are not significant as the observed value of r = ……………. is higher / lower than the critical value …………., where N=………………… for a ……………. -tailed hypothesis, with a p ≤ …………. level of significance, the null hypothesis can be rejected / accepted, and the alternative hypothesis rejected / accepted.

Tests of Correlation: Spearman's Rho

Spearman's test is a test of correlation between two sets of values. This test is selected when both variables are ordinal level. The type of design is not relevant here as the investigation is correlational rather than experimental.

Step 1: state the hypotheses

Step 2: rank each set of scores separately in each group/condition from lowest to highest. If two or more scores share the same ranks, find the mean of their total ranks.

Step 3: compare the calculated value with the critical value. For Spearman's test, the calculated value must be equal to or exceed the critical value for there to be significance.

Step 4: write a statement of significance.

The results are / are not significant as the observed value (calculated value) of rs = ……………. is higher / lower than the critical value …... where N=………………… for a ……………. -tailed hypothesis, with a p ≤ …………. Level of significance, the null hypothesis can be rejected / accepted, and the alternative hypothesis rejected / accepted.

Tests of Association: Chi-Squared

A test for a correlation or difference between two variables or two conditions. Data should be nominal level (or better) using an unrelated (independent) design.

Step 1: state the hypotheses

Step 2: place raw data in a contingency table

Step 3: find observed value of $X^2$ and compare with critical value ($df$ value = (no. of rows – 1) x (no. of columns – 1))

For the Chi-squared test, the calculated value must be equal to or exceed the critical value for there to be significance.

The results are / are not significant as the observed value (calculated value) of $X^2$ = ……………. is higher / lower than the critical value …... where $df$ value =………………… for a ……………. -tailed hypothesis, with a p ≤ …………. Level of significance, the null hypothesis can be rejected / accepted, and the alternative hypothesis rejected / accepted.