Analysis of Variance (ANOVA) and t test (Parametric tests)
Graham Tall G.E.Tall@bham.ac.uk 2003
Reference:
Ferguson, G.A. & Takane, Y (1989) Statistical Analysis in Psychology and
Education (Sixth Edition) London:McGraw-Hill
The purpose of studying these tests is to consider the tasks they can tackle and concentrate on how the tests are used and interpreted. The tests described here are more precise than ordinal and nominal tests but are simpler than other parametric tests. Hence, study the designs given in regression analysis to see if you can build the extra requirements into your research - it could make a critical difference to the end result.
i. Comparing groups using one piece of data (mark) per
individual
ii. Comparing a
single group of students which have been given the same test a number of
times.
iii. Test Information and Calculations
i. Comparing groups using one piece of data (mark) per individual:
Background: Consider the diagrams below:
In diagrams a) and b) the mean scores of the two groups differ. A problem arises in diagrams c) & d) because Analysis of Variance and the t test both assume that the distribution is approximately normal (when the scores are plotted on a graph they are 'bell-shaped', as in the diagrams above) and that the standard deviations of the groups are comparable; if they are not comparable then this, rather than the mean difference, could cause rejection of the null hypothesis - in such a case use an ordinal test to compare the two groups.
Parametric tests require interval data (e.g. marks) and All parametric tests referred to here, except F test, are two-tailed.
Example: Are there any differences in the scores of the groups of students below?
| Scores of individuals in each group. | Group 1 |
Group 2 |
Group 3 |
|
Mean |
20.6 |
21.9 |
28.0 |
|
ANOVA: F = |
3.976 |
|||
t test: t = |
Test can only be used with two groups of data. |
|||
F Ratio Test:
F= |
1.039 4 Not Sig. |
4.265 6 Sig @ 5% |
2.093 11*** Not Sig. |
|
| Note *: | df = degrees of freedom |
| Note **: | To discover if F or t is significant need to use tables usually found at the back of all statistics books. In the tables the first degree of freedom identifies the column and the second identifies the row. |
| Note ***: | The row df is 11 rather than 5 because the final step in calculating the F ratio in this test requires the larger variance to be divided by the smaller, the F score found by the test must be greater than 1. The F ratio of the test is therefore different from the F ratio’s discovered using analysis of variance and the F tables have to be used differently; the F representing a one-tailed result. However since the F tables are designed for the standard analysis of variance two-tailed tests the table headed 5% is equivalent to the F tests one-tailed result. |
| Note: | Unlike Chi square it makes no difference whether raw or percentage scores are used and, it is perfectly legitimate to make all of the scores smaller e.g. subtract 30 if it make the sums easier.) |
Tests Used:
Analysis of Variance (ANOVA
- 2+ groups) or t test (2 groups) - compares the means
of the groups. Note: when there are two groups the result of the two tests is
identical, F=t2. When there are more than two
groups, the test has discovered that the samples do NOT all come from the same population
it has NOT identified which groups differ
The advantage of ANOVA over repeatedly using the t test, when there are more than 2 groups, is that "if you have 5 groups and compare all the pairs of means, you’re making 10 comparisons. When the null hypothesis is true, the probability is that at least one of the 10 observed significance levels will be less than 0.05 is about 0.29. The more comparisons you make, the more likely it is that you’ll find one or more pairs to be statistically different, even if all population means are equal." Norusis, M.J. (1992) SPSS for Windows Base System User’s Guide Release 5.0. SPSS:Chicagop.265. In SPSS, request Bonferroni test with ANOVA to discover which means differ significantly from the others (Norusis p265)
F Ratio test (sometimes called variance ratio test) - this compares the variance (the spread of the scores of the two groups or, of one group to the remainder). Strictly, this test should always be done before the Analysis of Variance or t test. In the example above, the standard deviation in Group 2 is identified as different from that of the other two groups combined. This means that the analysis of variance test should only be used to compare groups 1 and 3 (with F=4.746, df 1 & 9 the difference would not be significant), to compare group 2 and 3 use an ordinal test. If there are just two groups and the F test indicates that they have different standard deviations, then the average scores can be compared if the data is rank ordered, using the Mann-Whitney U test. Top of the Page
ii. Comparing a
single group of students which have been given the same test a number of
times.
These comparisons require a repeated measures version of
Analysis of variance or t test.
iii. Test Information and Calculations
These tests will be referred to by most if not all books on
statistics:
Guilford, J.P. (1965) Fundamental Statistics in Psychology and Education. McGraw-Hill:
London (discusses F ratio test)
Ferguson G.A. & Takane, Y. (1989) Statistical Analysis in
Psychology and Education. McGraw-Hill: London
It is not expected that you will
normally need to work through the
formulae in such books. Computer programs are readily available. Recommend:
SPSS, Minitab
or
EXCEL/WORKS. Spreadsheets containing
the necessary formulae can be bought from Graham Tall. The analysis of
variance (ANOVA) spreadsheet carries out repeated measures and different group
tests using ANOVA and the t test. The spreadsheet automatically checks
whether the standard deviations (variances) allow the use of the test.
Ordinal tests should be used if the variances differ statistically.
.Note: In SPSS & Mintab the data must be in a single file with the groups identified by a single variable eg: Variable=School with School A=1, School B=2, School C=3 etc.
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