Parametric Tests
Basic
Parametric tests are more power efficient (better at detecting genuine differences) but there are restrictions on what data can be used.
Assumptions
- Interval level data
- homogeneity of variance
- normally distributed level of data
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Simple Tests of Difference
| Differences or Correlation | Type of Design | |
|---|---|---|
| Related t-test: mean of differences between pairs of related value. | Differences | related (repeated measures/matched pairs) |
| Unrelated t-test: difference between two means of two sets of unrelated values | differences | independent |
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More than 2 Conditions
One Way ANOVA (Analysis of Variance)
One Way ANOVA: tests the null hypothesis that two or more samples were drawn from the same population by comparing means
compares the between-group variation with the within-group variation
| Differences or Correlations | Type of Design | |
|---|---|---|
| Unrelated ANOVA | Differences | between groups/subjects (independent) |
| Related ANOVA | Differences | within groups/subjects (repeated measures) |
| Mixed ANOVA | Differences | between and within groups |
Interpreting ANOVA Example: reject the null hypothesis that the sample means are of groups with identical population means
| Source of variation | Sum of Squares | df | Mean sum of squares | F ratio | Probability of F |
|---|---|---|---|---|---|
| between groups | 45.17 | 2 | 22.59 | 4.446 | p<0.05 |
| error | 45.75 | 9 | 5.08 | ||
| total | 90.92 | 11 |
Interpreting the F result: A priori and Post hoc comparisons
Post hoc comparisons: comparisons made after inspecting the results of the ANOVA
A priori comparisons: comparisons we can make, having made a specific prediction, based o theoretical argument, before conducting ANOVA
post hoc comparisons produce a much higher probability of making a Type 1 error than occurs with a priori
| A Priori Tests | Post hoc Tests |
|---|---|
| Bonferroni t tests - several planned comparisons | Newman-Keuls test - all possible pairs of means |
| linear contrasts - one or two planned comparisons between pairs of means/combinations of groups | Tukey’s (honestly significant difference) test = all possible pairs of means where there are 5+ groups |
Multivariate ANOVA
each independent variable is known as a factor and each of these have several levels
if all the factors of a complex ANOVA design are between groups (independent samples for each level) then it is an unrelated design
if all participants undergo all combinations of conditions (appear in every cell of the table), it is a repeated measures design
if at least one of the factors is unrelated and at least one repeat measure, then it is a mixed design
Main Effect: occurs when onne of the IVs has an overall significant effect
Simple Effect: when we extract part of a multi-factor ANOVA result and look at just the effect of one level of one IV across one of the other IVs
Interpreting multivariate ANOVA example. = neither IV had a significant effect in isolation (no main effect). However, there is a significant interaction effect (could prompt testing simple effect)
| Source of Variation | Sum of Squares | df | Mean sum of squares | F ratio | Probability of F |
|---|---|---|---|---|---|
| Between groups: IV 1 | NS | ||||
| Between groups: IV 2 | NS | ||||
| Between groups: Interaction (1x2) | p<0.01 | ||||
| Error | |||||
| Total |
Repeated measures ANOVA (one way or multi-factor)
Interpreting one-way repeated measures ANOVA example: hypothesis that significant difference between conditions is supported
| Source of Variation | Sum of Squares | df | Mean sum of squares | F ratio | Probability of F |
|---|---|---|---|---|---|
| between subjects | |||||
| within subjects | |||||
| between conditions | p<0.001 | ||||
| Error | |||||
| Total |
Interpreting two-way repeated measures ANOVA example: significant main effects of both factors and significant interaction effect
| Source of Variation | Sum of Squares | df | Mean sum of squares | F ratio | Probability of F |
|---|---|---|---|---|---|
| <<Factor A<< | p<0.02 | ||||
| Factor B | p<0.02 | ||||
| Interaction | p<0.05 | ||||
| Error | |||||
| Total |
Interpreting two-way mixed modal (2 unrelated, 3 related) ANOVA example: IV 2 had an effect but it is limited to 1.1 only
| Source of Variation | Sum of Squares | df | Mean sum of squares | F ratio | Probability of F |
|---|---|---|---|---|---|
| <<Between subjects (IV 1.1 and IV 1.2)<< | NS | ||||
| within subjects (IV 2) | p<0.01 | ||||
| within subjects (interaction 1.1/1.2x2) | p<0.001 | ||||
| Error | |||||
| Total |
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