Please enable JavaScript.
Coggle requires JavaScript to display documents.
Analysis of Variance (ANOVA), ANOVA t statistic: Screenshot 2025-11-16 at…
Analysis of Variance (ANOVA)
if there is variance between groups there is significant effect, if there is variance within groups there is not significant effect
is a hypothesis-testing procedure that evaluates mean differences three or more treatments or populations.
ANOVA is different from t-test, as t-test can only test two treaments or populations.
In ANOVA, factor refers to the variable (independent or quasi-independent) that designates the groups being compared.
Levels are the individual conditions or values that make up factors.
ANOVA can use independent and repeated measures design.
Two-factor design or a factorial design: a study that combines two factors.
Independent-meausures designs: study that use separate group of participants for each treatment condition
Null Hypothesis:
Single-factor designs: studies that only have independent variables or only one quad-independent variable.
Alternative Hypothesis: can also indicate there is a significant difference between a specific treatment/population but not another.
Type I errors differ in ANOVA, as the risk of type I errors increase the more test are done.
Testwise alpha level: is the risk type I errors or alpha level for an individual hypothesis test.
Experimentwise alpha level: is when an experiment involves several different hypothesis test that is the total probability of a type 1 error that is accumulated from all individual test in an experiment.
ANOVA uses one test with one alpha level to evaluate the mean differences, thereby avoiding the problem of an inflated experiment wise alpha level
ANOVA t statistic:
Possible differences between samples:
1-differenced are naturally occurring and are a result of sampling error.
2-treatment effects are the causation between treatments differences. Therefore, scores in one treatment should be systematically different from scores in another condition.
To measure random and unsystematic factors, we compute variance within treatments.
Within treatment variance, the differences that exist within a treatment represent random and unsystematic differences that occur when there are no treatment effects causing the scores to be different.
Error term: is a measure of variance caused by random and unsystematic differences. When the treatment effect is zero (Ho 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.
9-steps required to calculate for the ANOVA:
Find the three values for SS and df, two variances for between and within treatments, and lastly find the F-ration
Total Sum of Squares using the computational formula:
This is the formula consistent with ANOVA:
Between and within Sum of Squares:
Alternative formula for between treatment that only works if there are whole numbers and same size sample:
Total Degrees of freedom:
Within degrees of freedom:
Between degrees of freedom:
ANOVA statistics continued...
Mean square: is used in place of variance.
Between:
Within:
false null hypothesis f-ratio should be much greater than 1.00.
for a true null hypothesis, the numerator and denominator of the f-ratio are measuring the same variance and the two samples should be around the same size for the ratio to be near 1.
f values are always positive. So they will pile around 1.00 or taper off to the right.
Post hoc tests: Turkey's HSD test
effect size:
