One-Way Analysis of Variance (ANOVA)

*Between-groups Variance*

*Between-groups Variance* based on means within groups across all samples to estimate the variance. Variance based on mean of group 1> Variance based on mean of group 2> Variance based on mean of group 3> GRAND MEAN OF ALL SCORES IN STUDY *Between-groups Variance* (MS between): Averages of the group variance estimates in SPSS called MS "Error"

Decision for the ANOVA

*F ratio more extreme* than critical F value (larger) -- Reject null hypothesis of no difference between the means -- Support research hypothesis *F ration less extreme* than critical F value (smaller) -- Retain null hypothesis of no difference between the means -- no evidence to support research hypothesis

Interpreting the F Ration

*Null hypothesis*: No difference between population means (μ1=μ2=μ3) --MS between roughly equal MS within; F is close to zero (F=1 or not significantly *different from 1*) *Research Hypothesis*: Population means are different -- MS between > MS within; F is greater than 1 (F> 1 by a significant amount) Difference between groups. Compare the F ration to the critical value just like the t test.

Two Sources of Variability in the ANOVA

*Within-groups Variance* based on individual scores within groups. (Score, Score,Score,Score,Score,) *Between-groups Variance* based on differences between group means: <>Group 1 <> Group 2 <>Group 3<>

Degrees of Freedom in ANOVA

1) BETWEEN: *-df between = (N groups -1)* 2) WITHIN: df within = sums of df's of each individual group df of the group = N-1 of group, then add these together for each group: *(Df1+df2+...df last) * 3) DEGREES OF FREEDOM TOTAL: df total = (df between +df within)

Present the Results

A one-way ANOVA was conducted to determine whether observing helping behaviors affected later occurrences of helping behavior. The results were significant, F(2,21)= 5.73, p< .05. Based on these results, we can reject the null hypothesis that there is no difference between the mean scores of the groups. Post hoc tests reveal that both the video group (M= 2.25, SD =x) and the live observation group (M=2.38, SD =x) exhibited significantly more helping behaviors than the natural conversation group (M=1, SD =x).

One-way *Between-groups* ANOVA

Example: Helpfulness study examines the rates of helpful behavior after participants are exposed to one of three conditions: 1. Watching a video where one person helps another. 2. Observing one person help another in a live situation. 3. Having a neutral conversation with a researcher (Control Group). *| LEVEL 1 | LEVEL 2 | LEVEL 3 |* Data set includes CONTROL: Neutral conversation Tx 1: Video helping Tx 2: Live helping *SPSS Reminder*: Data will be entered differently in SPSS. Create a *grouping variable for the IV* in Column 1. Enter *score variable for DV* for the DV in column 2.

The F Statistic

F= Ratio of between-groups variance ("mean square" between) to within-groups variance ("mean square within") f = MS between/MS within Between groups variance is what we are interested in-it represents the difference between groups due to the intervention. Not due to random error or individual differences. *DESIRED RESULTS*: MS between > MS within * Not due to random error or individual differences.

*Within-groups Variance*

POPULATION VARIANCE: treats scores separately. More than 2-3 groups. *Within-groups Variance* based on individual scores within groups. (Score, Score,Score,Score,Score,) Variance based on scores of group 1> Variance based on scores of group 2> Variance based on scores of group 3> *Within-groups Variance* (MS within): Averages of the group variance estimates in SPSS called MS "Error"

Effect Size in ANOVA

R^2 represents the proportion of overall variance that is accompanied for by the independent variable (that is, differences between groups) FORMULA: R^2 SS between/SS total If R^2 = .35, then 35% of the overall variance is accounted for by the IV or effect of the intervention. This is a large effect. In SPSS: effect size is called *eta squared* ( see tutorial)

Calculations

You won't have to calculate entire ANOVA's by hand, you need to understand how to calculate a few crucial values when given other values: -- 3 types of degrees of freedom (df) -- F Statistic -- R^2 (effect size measure) * SPSS uses eta squared n^2 - interpreted the same way.

One-Way Within groups ANOVA

ANOVA EXAMPLE: Same study, but all participants are assigned to all groups, or all levels of the independent variable. They will experience each condition at different times. Aside from eliminating one type of variance, the within-groups ANOVA is calculated the same as the between-groups ANOVA. <>Group 1 <> Group 2 <>Group 3<>

ANOVA

DV is scale IV is nominal or ordinal Has at least 3 groups or 'levels" -- Independent t test: IV Type of treatment has only two levels: Tx 1, Tx 2 -- One-Way ANOVA: IV Type of treatment has 3 or more levels: Tx 1, Tx 2, Tx 3

Two Types of ONE WAY ANOVA's

HOMEWORK FOCUSES ON THIS: One-way *Between-groups* ANOVA - each participant is in only one group, or level, of the IV (Similar to independent-samples t test, but with more than 2 groups). One-way *Within-groups* ANOVA - each participant experiences every level of the IV (Similar to paired-samples t test, but with more than 2 groups).

When to Use ANOVA analysis of variance

Independent-Samples t test would require there different tests such as control and treatment with outcome on dependent variable. Increases the chance of Type 1 error quite possible. Sample and population means. ONE WAY ANOVA: Control, tx 1, tx 2 with the Outcome on Dependent Variable. This looks at the variations of the groups. So the variances show the differences between groups.

The F Table

Used to look up the cut-off F value for a study. F distribution starts at ) and increases - F rations are always positive. Similar to t-table, but have to take into account 2 types o degrees of freedom. *-df between = Ngroups -1* *-df within = (Df1+df2+...dflast) * Compare f ration from study to value in F table.

Post Hoc Tests

When F is significant, we know there is some difference between groups, but not which groups, Post Hoc (after the fact) tests make group-to-group comparisons to determine which groups are significantly different from others. *YOU MUST RUN post hoc analyses if F is significant.* SPSS Examples: Turkey's HSD, Fisher's LSD>> OMNIBUS TEST