8A Class Slides - One Way ANOVA PART 1b

PSYCH 200: ANOVAPART 1B : 

PSYCH 200: ANOVAPART 1B Decomposing variance One-way ANOVA Multiple comparisons

One Way ANOVA Example : 

One Way ANOVA Example Imagine you are performing a study in which you are interested in the effect of magnetism on moral reasoning. You believe that a magnetic wave pointed at a certain part of the brain can affect our moral decision making.You have 16 people come into the lab. 5 of them are in the control condition and are not exposed to any magnetic wave (control 1), 6 are in the magnetic wave condition at the part of the brain responsible for moral decision making (experimental condition), and 5 are also exposed to a magnetic wave, but at a part of the brain not responsible for moral reasoning (control 2).After the manipulation, everyone takes a test of moral reasoning on a scale of 1-10.

One Way ANOVA Example : 

One Way ANOVA Example H1: Magnetic Waves can affect moral reasoning μcontrol 1, μcontrol 2, μexperimental, are not all equal H0: Magnetic Waves cannot affect moral reasoning μcontrol 1, μcontrol 2, μexperimental, are all equal Factor ? Magnetic Wave Level Levels ? 3: Control 1, Control 2, Experimental DV ? Moral Reasoning Test (1-10 scale) STEP 1: Null and Alternative Hypotheses STEP 2: Identity Factor, Levels, and DV

Example : 

Example = 6.38

Slide 5: 

3 4 5 6 7 8 Control 2 Control 1 Experimental Example

Decomposing variance : 

Decomposing variance “Natural variability” “Variability across group means” F = “Estimate of population variance” “Average deviation from grand mean”

Decomposing variance : 

Decomposing variance STEP 3: We need to identify the two sources of variance (Between and Within/Natural/Error) We need equations to do that… Well… let’s think about what variance is.

Decomposing variance : 

F = “Average deviation from grand mean” “Estimate of population variance” General formula for variance of a set of numbers: SS df MSB MSW Decomposing variance

Variance within-groups : 

Variance within-groups a.k.a. Natural Variability or Error variance As always, we are trying to obtain the best estimate of the (common) population variance, σ 2 Recall the independent-samples t, where we pooled the variance across samples to estimate σ 2 Similarly, because we also assume homogeneity of variance in the ANOVA, we use a pooled estimate So what is that pooled estimate equation?

Variance within-groups : 

Variance within-groups a.k.a. Natural Variability or Error variance A bit of notation first… Xi,j refers to the some score X in group J Xj refers to the average of group J

Variance within-groups : 

Variance within-groups Mean squared error (or within-groups), MSW SS df N - 1 N - k MSW = SS1 + SS2 + … + SSk

Slide 12: 

Variance within-groups Mean squared error (or within-groups), MSW 3 4 5 6 7 8 Control 2 Control 1 Experimental

Variance within-groups : 

Variance within-groups Mean squared error (or within-groups), MSW So the equation is what?!? Well, the equation for MSw (pooled variance) for a One Way Between Subjects ANOVA is… Σ (Xi,j – Xj ) N - k MSW = Sum of Squares Within (SSw) Degrees of Freedom Within (dfw) k = number of groups/levels in IV 2

Back to our Example… : 

Back to our Example… = 6.38

Back to our Example… : 

Back to our Example… = 6.38 = 13.20 2

Back to our Example… : 

Back to our Example… SSw = 13.20 dfw = N-k Well, in our example, we had an N of 16. And we had 3 groups in our IV (control 1, control 2, experimental) So our dfw is 16 - 3 = 13

Decomposing variance : 

General formula for variance of a set of numbers: SS df MSB MSW Decomposing variance MSW = SSw/dfw MSW = 13.20/13 = 1.105 Next step… we need to find MSB (Mean Square Between)

This is the end of Part 1B : 

This is the end of Part 1B