Post-hoc Tests for ANOVA

post hoc analyses are usually concerned with finding patterns and/or relationships between subgroups of sampled populationthat would otherwise remain undetected and undiscovered were a scientific community to rely strictly upon a priori statistical methods

Tests

•Fisher's least significant difference (LSD)

•Bonferroni procedure

•Holm–Bonferroni method

•Newman–Keuls method

•Duncan's new multiple range test (MRT)

•Rodger's method

•Scheffé's method

•Tukey's procedure

•Dunnett's correction

•Sidák's inequality

•Benjamini–Hochberg (BH) procedure

Post hoc procedures vs. a priori comparisons (we look at only post hoc)

Comparisonwise error rate - alpha for each comparison

Experimentwise error rate - the probability of making a Type I error for the set of all possible comparisons

alphae = 1 - (1-alpha)c

SPSS One-Way ANOVA with Post Hoc Tests

A hospital wants to know how a homeopathic medicine for depression performs in comparison

to alternatives. They adminstered 4 treatments to 100 patients for 2 weeks and then measured

their depression levels. The data, part of which are shown above, are in depression.sav.

ANOVA - Main Assumptions

• Independent observations often holds if each case (row of cells in SPSS) represents a unique person or other statistical unit. That is, we usually don't want more than one row of data for one person, which holds for our data;
• Normally distributed variables in the population seems reasonable if we look at the histograms we inspected earlier. Besideds, violation of the normality assumption is no real issue for larger sample sizes due to the central limit theorem.
• Homogeneity means that the population variances of BDI in each medicine group are all equal, reflected in roughly equal sample variances. Again, our split histogram suggests this is the case but we'll try and confirm this by including Levene's test when running our ANOVA.

There's many ways to run the exact same ANOVA in SPSS. Today, we'll go for General Linear 

Modelbecause it'll provide us with partial ate square as an estimate for the effect size of our 

model

. We'll briefly jump into Post Hoc and Options before pasting our syntax.

The post hoc test we'll run is Tukey’s HSD (Honestly Significant Difference), denoted as 

“Tukey”. We'll explain how it works when we'll discuss the output.

“Estimates of effect size” refers to partial ate square. “Homogeneity tests” includes Levene’s 
test for equal variances in our output.

SPSS ANOVA Output - Levene’s Test

 Levene’s Test checks whether the population variances of BDI for the four medicine groups 
are all equal, which is a requirement for ANOVA. “Sig.” = 0.949 so there's a 94.9% probabilityof
finding the slightly different variances that we see in our sample. This sample outcome is very
likely under the null hypothesis of homoscedasticity; we satisfy this assumption for our ANOVA.

SPSS ANOVA Output - Between Subjects Effects

 If our population means are really equal, there's a 0% chance of finding the sample

differences we observed. We reject the null hypothesis of equal population means.

 The different medicines administered account for some 39% of the variance in the BDI

scores. This is the effect size as indicated by partial eta squared.

Partial Eta Squared is  the Sums of Squares for medicine divided by  the corrected total

sums of squares (2780 / 7071 = 0.39).

 Sums of Squares Error represents the variance in BDI scores not accounted for by medicine. Note that  +  = .

comparing 4 means results in (4 - 1) x 4 x 0.5 = 6 distinct comparisons, each of which is listed

twice in this table. There's three ways for telling which means are likely to be different:

 Statistically significant mean differences are flagged with an asterisk (*). For instance, the

very first line tells us that “None” has a mean BDI score of 6.7 points higher than the placebo 

which is quite a lot actually since BDI scores can range from 0 through 63.

 As a rule of thumb, “Sig.” < 0.05 indicates a statistically significant difference between

two means.

 A confidence interval not including zero means that a zero difference between these means

in the population is unlikely.

Obviously, ,  and  result in the same conclusions.

by : Fatin Farhana bt Marzuki

References

https://www.spss-tutorials.com/spss-one-way-anova-with-post-hoc-tests-example/

https://statistics.laerd.com/statistical-guides/one-way-anova-statistical-guide-4.php