ANOVA notes - PowerPoint Presentation

Slide1

ANOVA notes

NR 245

Austin Troy

Based primarily on material accessed from Garson, G. David 2010.

Univariate GLM, ANOVA, and ANCOVA.

Statnotes

: Topics in Multivariate Analysis.

http://faculty.chass.ncsu.edu/garson/PA765/statnote.htmSlide2

Central tendency refresher

Mean

Median

VarianceStandard Deviation

For sample

Variance for populationSlide3

ANOVA

“

main effect”

vs interactive effects of categorical independent variables (factors) on a continuous/interval dependentTests for overall differences in means, not variances

Pairwise comparisons Slide4

Whether difference exists depends

on:

Size of differences between group means.

Sample sizes in each group. Variances of dependent variable by

groupSlide5

Fixed vs. random effects ANOVA

Fixed: data

are collected on all categories of independent variables.

Factors with all category values included are "fixed factors." Random

effect ("Model II"), data collected only for a partial sample of categories.One-way ANOVA: computation of F is the same for fixed and random effects

Mixed effects models with both typesSlide6

Research design: between groups

Dependent variable is measured for independent groups of sample members, where groups represent different condition, or categories.

Experimental mode

: conditions assigned randomly to

subjects, or subjects assigned randomly to conditionsNon-experimental

mode, conditions are measures of the independent variable for each group.Slide7

Full factorial ANOVA

More than one factor: two-way or higher

In this approach, each cell becomes a “group”

Source: http://faculty.chass.ncsu.edu/garson/PA765/anova.htmSlide8

ANOVA assumptions

Interval data

.

nonparametric Kruskal-Wallace Homogeneity of variances.

Box plots Multivariate normality. Slide9

ANOVA assumption

Adequate sample size

.

Random samplingEqual or similar sample sizes. Slide10

Interpretting

2+way

ANOVA

Profile plots

Color= 2nd

factor (e.g. gender)Parallel lines= lack of interaction effectsSeparate lines = different means based on genderTriangle or X= different groups means based on region

Bottom row: both effects plus interaction

Source: http://faculty.chass.ncsu.edu/garson/PA765/anova.htmSlide11

F test for comparing group means

For most designs, F is between-groups mean square variance divided by within-groups MSV

If F >1

, then there is more variation between groups than within groups, = the grouping variable does make a difference. Significance of F stat: using

df=k-1 (between group; df for numerator) and df=N-k-1 (within group;

df for denominator)Larger the ratio of between-groups variance (signal) to within-groups variance (noise), the less likely that the null hypothesis is

true=more variation between groups than within groupsSlide12

Pairwise comparisons

Assess group differences

The

possible number of comparisons is k(k-1)/2. For two comparisons use standard t

testFor more comparisons:Bonferroni test:

like t-test but adjust the significance level by multiplying by the number of comparisons being made. Tukey’s HSD test: like a t-test,

but corrects for experiment-wise error rate; gives conservative results when group sizes are

unequal; good

with large number of categories.

Slide13

ANOVA outputs

Example: pain level by analgesic drug group

Rsquare

is model of goodness of fit and is often referred to as “partial eta squared” for ANOVA= ratio of the between-groups sum of squares to the total sum of squaresAdjusted R-square adjusts the

Rsquare value to penalize for more parameters by using the degrees of freedom in its computation R-adj= 1 - SSE(n

-1)/SST(v)Root MSE gives the standard deviation of the random errorMean of response gives sample meanSlide14

ANOVA outputs

This section gives F test results

DF= degrees of freedom

Sum of Squares C. Total= sum of squared distances of each response from the sample mean; Error is the residual or unexplained SS after fitting the model Mean square is a sum of squares divided by its associated

dfF score is ratio of the Model mean square to the error mean square

Prob>F is p value; observed significance probability of obtaining a greater F-value by chance alone. 0.05 or less considered evidence of a regression effect.

Also get Mean, Std Error (in this case is the root mean square error divided by square root of the number of values used to compute the group mean. and confidence intervals for each groupSlide15

ANOVA outputs-comparisons

Top: LSD

Threshold

matrix: absolute difference in the means minus the LSD, which is the minimum difference

that would be significant. Pairs with a positive value are significantly different. The q statistic is a scaling variableNext table: significant differences based

on the letters that apply to them (A, B)Final table gives all pairwise comparisons. Gives differences, confidence intervals and p value. Those with significant differences are starredAlso gives diamond and circle plots. If two circles overlap significantly, those groups are not differentSlide16

Box plot

Outliers

are either 3×

inter-quartile range (

the width of the box in the box-and-whisker plot) or more above the third quartile or 3×

IQR

or more below the first quartile