11 Anova - Patrick Harrison PDF

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Patrick Harrison...

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ANOVA Monday, April 2, 2018

2:39 PM

Introduction: - T-tests are limited - More than two means? Use ANOVA - Concept of hypothesis testing with the partitioning variance Example: - How to teach statistics? - Three techniques ○ Just lecture ○ Just textbook ○ Combination of both - Would be Multiple t- tests - Do the three conditions differ from each other? ○ What ANOVA answers - If so, how? General Model for ANOVA - Goes along with regression - Mu - grand mean - Ti - group mean (j) minus grand mean ○ Deviation of condition mean from the grand mean - Eij - error ○ Deviation of participant I from condition mean Its all about the variance - Total sum of squares - if we divided this by the degrees of freedom (N-1) we would get the total variance RECAP - More than 2 means? Use ANOVA - Breaks down total variance in two parts - Between groups variance - Within groups variance - SS total =SS between + SS within ○ SS = sums of squares

○ SS sums of squares ○ additive - Divide sums of squares by their degrees of freedom ○ We don’t like sums because they don’t tell you much - Outcome: mean squares (MS) (like an average) ○ No longer additive so compare - Compare MSbetween to Mswithin - F = MS between/MS within ○ If F is large enough you can make judgments about the hypotheses So what - F ratio tells us how important group differences are relative to how much groups differ within themselves ○ Homogenous group but one differs = significant F - Higher MS between than MS diff = larger F ANOVA model - H0: M1 =M2=M3 ○ Null hypothesis says that all means came from the same population ○ If null is true, we just have error variance - H1: not H0 ○ Says that at least one of the means came from a different distribution ○ If we reject the null hypothesis, it means that some of this variance is due to the differences among the conditions ANOVA model - So when the null hypothesis is true: ○ MS between = MS within ○ F=1 - When the null hypothesis is false: ○ MS between is larger than MS within ○ F>1 F distribution - For Z and t distributions, the sampling distribution of the mean is normal - Variance do not work this way - There are no negative values and they don’t distribute themselves normally ○ This is because variance are always squared - MS within comes up on SPSS as error variance - If greater than critical value, reject null ANOVA and F

- Uses F to test hypotheses - If all samples are drawn from the same distribution, F should be near 1.0 - If one or more samples come from different distributions, F gets larger - If F gets large enough, we reject the null hypothesis Summary - Partition variance in two parts - Null hypothesis: MS between and MS within are just error variance - Null hypothesis is false: - MS between = group differences+ error - MS within = error Which means differ? - Planned contrasts (a priori) ○ Use with one-tailed test because you know what you are looking for - Post hoc tests Post Hoc Tests - Fisher's LSD test (least significant different) ○ Is there anything possibly going on here? Very sensitive ○ More likely to find false positives - Tukey Test ○ More conservative - Scheffe Test - Bonferroni Correction All possible comparisons - See if any means differ - Increases the chance of a type 1 error - For example if we did 6 comparisons… P of type 1 error = 26% Summary - More than two means? Use ANOVA - ANOVA partitions variance - Is an omnibus test - just overall group difference...