Understanding and conducting ANOVA Understanding and conducting

Basics of ANOVA Null hypothesis: All groups are coming from the same population Differences in means are due to normal sampling fluctuations between samples drawn from the same population If null hypothesis is true, then certain other things must be true.  Test if that is the case (F-test) If yes  accept null hypothesis If no  reject null hypothesis

Example Students were grouped into low-medium and high levels of motivation. They were asked number of hours per week they spent doing homework. The question is, whether low-medium-high motivated students spend time doing homework that significantly differs.

Example Null and Alternative Hypotheses

Example Means of each group

Example Under the null hypothesis, all of these are coming from the same population Variance in means

Example Between groups estimate of variance

Example Within groups estimate of variance

F-Statistic (Fisher’s test, 1923) Logic of the F statistic: The two estimates would be approximately equal if samples were drawn randomly from a single population. If the ratio of “between” samples variance to “within” samples variance is taken, it should be approximately 1. Depending on sample size Depending on variation of sample means among all possible samples If the ratio of “between” samples variance is much higher than “within” samples variance, then sample means vary more than expected by chance. This is evidence that the independent variable is associated with significant differentiation of means.

Understanding the F-test Between groups variance Within groups Variance due to membership of groups True variance between subjects measurement error + =

Assumptions of the F-test Interval/ratio level measurement of the dependent variable (robust against deviations from normality) Independence of observations (each unit of analysis is independently selected into the sample)

Which one is NOT a null hypothesis for ANOVA comparing the political conservatism of low, middle and high SES individuals? Low, middle, and high ability groups do not differ from each other. Low, middle, and high ability groups are just like simple random samples drawn from the same population. Low, middle and high ability groups differ from each other in their political conservatism. SES of individuals is not associated with their political conservatism.

How is an F test differ from a t test? T/F - The two tests are different because their null hypotheses are different.

How is an F test differ from a t test? T/F - The two tests are different because the t test compares 2 group means and the F test can compare multiple group means.

How is an F test differ from a t test? T/F - The two tests differ in the way we interpret the p values.

How is an F test differ from a t test? T/F - The two tests differ in the way we reason when we make inferences from a sample to a population.

The F test works by comparing two estimates The F test works by comparing two estimates. Those two things are the estimates of WHAT? Population means Population variances Sample means Sample standard deviations

There are two different estimates of the population variance in an F test. One is called the WITHIN GROUPS estimate of the population variance. Which statement is not true about the WITHIN GROUPS variance? WG variance is given that name because it is obtained by averaging person-to-person variance estimates for each group. WG variance is the variance that quantifies the person-to-person variability of all persons in all groups combined. WG variance is compared to the “between groups” estimate of the population variance for an F test. WG variance is just a summary estimate of what the population variance is, based on each one of the sample variances.

One of the estimates of the population variance in an F test is called the BETWEEN GROUPS estimate of the population variance. Which statement is not true about the BETWEEN GROUPS variance? BG variance is given that name because it is obtained by focusing on the variability of group means. BG variance is the variance estimate that uses the SEM formula relating the population variance to the sample-to-sample variability. BG variance is compared to the “within groups” estimate of the population variance for an F test. BG variance is the estimate of the variance under the assumption that the alternative hypothesis that the groups are coming from different populations.

When will the F test be equal to 1? When the differences between the sample means are large enough. When the alternative hypothesis is true. When the error variance is too large. When the null hypothesis is true.

Thinking exercise Spider web on ANOVA

Multifactorial Designs: Beginning Multivariate Analysis Read G&W 428-435

Excel file – Multifactorial ANOVA introduction More than one IW may influence a DV Sex Relationship Satisfaction Family Type

Interaction effects There can be many different types of interactions Interactions must be interpreted in the light of theory They are symmetric mathematically Main effects cannot be interpreted if there are interaction effects in the model Main effects are “marginal” to the interaction effects

Satisfaction is influenced by family background only in one group Intact Disrupted

Excel file – Multifactorial ANOVA introduction More than one IW may influence a DV Sex Relationship Satisfaction Family Type

Satisfaction is influenced by family background only in one group Intact Disrupted

Excel file – Multifactorial ANOVA introduction More than one IW may influence a DV Sex Relationship Satisfaction Family Type

Satisfaction is influenced by family background only in one group Intact Disrupted

Excel file – Multifactorial ANOVA introduction More than one IW may influence a DV Sex Relationship Satisfaction Family Type

Satisfaction is influenced by family background only in one group Intact Disrupted

SPSS output: Family type effects

SPSS output: Sex effects

SPSS output: Family type and sex effects at the same time

SPSS output: Family type and sex effects at the same time and interdependent

SPSS output: Family type and sex effects at the same time and interdependent

Understanding the F statistics: Equal sample sizes Factor B 1 2 … m 1 2 k n Factor A

Understanding the F statistics: Equal sample sizes Variance of the attribute in the subjects in each group Factor B 1 2 … m 1 2 k n Factor A

Understanding the F statistics: Equal sample sizes Factor B 1 2 … m 1 2 k n Factor A

Understanding the F statistics: Equal sample sizes Factor B 1 2 … m 1 2 k n Factor A

Understanding the F statistics: Equal sample sizes Factor B 1 2 … m 1 2 k n Factor A

Reviewing multifactor F statistics Within group variance Factor B 1 2 … m 1 2 k n Between groups variance of Factor A Factor A Between groups variance of Factors A*B Remember: This subtracts Factor A and Factor B marginal variances Between groups variance of Factor B

Language of Multifactorial ANOVA The present research has a m*n design. Main (marginal) effects of Factor A were significant (not significant). Main (marginal) effects of Factor B were significant (not significant). Interaction effects of Factor A by Factor B were significant (not significant).

READ AND STUDY: Example 14.2, page 429