Chapter 12: introduction to analysis of variance

An ANOVA finds significant treatment effects for a study comparing three treatments with means of M1= 8, M2= 4, M3= 2. If Tukey's HSD is computed to be HSD = 2.50, then which of the treatments are significantly different?

1 vs. 2 and 1 vs. 3

an analysis of variances produces df between treatments = 2 and df within treatments= 24. What is df total?

26

Effect size for ANOVA for independent measures

ANOVA is computed eta squared, the percent of variance accounted for by the treatment effect =SS between/SS between + SS within = SS between/ SS Total

MS within f ratio structure

F= treatment effect + differences due to chance/ differences due to chance

f ratio for independent measures anova

F=MS between/MS within

testwise alpha level

The risk of a Type I Error, or alpha level, for an individual hypothesis test.

ANOVA (analysis of variance)

a statistical technique that is used to test for mean differences among two or more treatment conditions. Null hypothesis for this test states in general population there are no mean differences among the treatments. The alternative states that at least one mean is different from another

df between treatments

df= k-1

researcher uses analysis of variance to test for mean differences among 4 treatments with sample of n=6 in each treatment. f-ratio for anaylsis would have what df values?

df=3,20

Within treatment df

dfwithin = E(n-1) = N-k

ANOVA F-ratio

f= variance (differences) between sample means/ variance (differences) expected with no treatment effect

f ratio in terms of its sources

f=systematic treatment effects + random, unsystematic differences/ random unsystematic differences

accurately describe the f ratio

is a ratio of two variances

The variances in the F ratio are called

mean squares, or MS values MS= SS/df

Formulas for ANOVA

p.326

MS within significant treatment effect

the numerator of the ratio should be larger than the denominator and the obtained f-value should be much greater than 1.00

differences of MS between treatments

treatment effects (if they exist) differences resulting from chance

f ratio has _______ values for degrees of freedom

two values 1. associated with MS in numerator 2. associated with MS in the denominator these df values are used to find critical value for f-ratio in f distribution table

how many levels are in a single factor independent measures design comparing depressions scores of participants with and without treatment?

2

Analysis of variance is used to evaluate the mean differences among three treatment conditions. Analysis produces SS within treatments = 20, SS between treatments = 40 and SS total= 60. For this analysis what is MS between treatments?

40/2

Research study compares three treatments with n=5 in each treatment. if SS valuers for the three treatments are 25, 20 and 15, then the analysis of variance would produce SS within equal to ________

60

measuring effect size. The percentage of variance accounted

= SS between treatments/ SS total

MS within when no treatment effect

H0 is true, the numerator and denominator of f-ratio are measuring the same variance, and the obtained ratio should be near 1.00.

between treatment sum of squares

SS between= SS total- SS within

within treatment sum of squares

SS within treatment= ESS inside each treatment

experimentwise alpha level

When an experiment involves several different hypothesis tests, is the total probability of a Type I error that is accumulated from all of the individual tests in the experiment. Typically, is substantially greater than the value of alpha used for any one of the individual tests.

post hoc tests

additional hypothesis tests that are done after an ANOVA to determine exactly which mean differences are significant and which are not

independent measures anova

analyzes two components: variance between treatments and variance within treatments

error terms

anova, the denominator of the f-ratio. Error term provides a measure of the variance caused by random, unsystematic differences. when the treatment effect is zero (H0 is true), the error term measures the same sources of variance as the numerator of the f-ratio, so the value of the f-ratio is expected to be nearly equal to 1.00

df total degrees of freedom

df total= N-1

levels

individual conditions or values that make up a factor

When decision from ANOVA is to reject the null hypothesis, and when experiment has more than two treatment conditions.....

it is necessary to continue the analysis with post hoc test, like HSD test or Scheffe test. Purpose of these tests is to determine exactly which treatments are significantly different and which aren't

MS between

measures differences between the treatments by computing the variability of the treatment means or totals. These differences are assumed to be produced by a. treatment effects (if existing) b. differences resulting from chance

MS within

measures variability inside each of the treatment conditions. individuals inside a treatment condition are all treated exactly the same, any differences within treatments cannot be caused by treatment effects. produces differences only caused by chance F= treatment effect and differences due to chance/ differences due to chance when no treatment effect (H0 is true), the numerator and denominator of f-ratio are measuring the same variance, and the obtained ratio should be near 1.00. if significant treatment effect, the numerator of the ratio should be larger than the denominator and the obtained f-value should be much greater than 1.00

Researcher uses an ANOVA to evaluate the mean difference between two treatment conditions and obtains F=4 with df= 1, 16. If independent measures t statistic had been used instead of ANOVA what would be obtained and what is the df value for t?

t=2.00 with df=16

F-ratio

test statistic for ANOVA is ratio of two variances.

factor

the variable (independent or quasi-independent) that designates the groups being compared

what is suggested by a large value for f-ratio in an ANOVA?

treatment effect and the null hypothesis should be rejected

for an analysis of variance, the differences between the sample means contribute to the __________ and appears in the ___________ of the f-ratio

variance between treatments, numerator

under what circumstances are post hoc tests necessary after an ANOVA?

when H0 is rejected and there are more than two treatments

when is the distinction between the "test wise" alpha level and the "experiment wise" alpha level important?

when the study is comparing more than two treatments