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Chapter 12: Analysis of Variance - Coggle Diagram
Chapter 12: Analysis of Variance
Analysis of Variance (ANOVA). A hypothesis testing procedures to analyze the difference of means among two or more populations.
Factor. The independent or quasi-independent being compared in an analysis.
Levels. The individual values or conditions that make up a factor.
Two factor or factorial design. A study that compares two factors.
Single factor design. A study that looks at one independent or quasi-independent variable.
The null hypothesis for a single factor design is Ho = µ1 = µ2 = µ3. The alternative hypothesis is there is a difference among means in the populations.
Test-wise alpha level. The risk of a Type 1 error for a single hypothesis
Experiment-wise alpha level. The risk of a Type 1 error for all of the hypothesis tests within an experiment. The experiment risk is higher than test-wise risk.
t statistic for ANOVA.Obtained difference between sample means / standard error.
F ratio for ANOVA. variance between sample means / variance expected with no treatment effect
When the F ratio is 1, then the treatment does not have an effect. When the F ratio is substantially larger than 1, then it does have an effect.
Error term. The denominator of the F ratio is called the error term, as it measures random and unsystematic differences.
To calculate ANOVA F ratio. Step 1: Obtain SS values between and within. Obtain df values between and within.
Step 2: Find variance between treatments SSbetween/df between and within treatments SSwithin/df within.
Step 3: Take variance between treatments / variance within treatments
F ratios are always positive, as variances are always positive.
When null hypothesis is true, the F ratio is measuring the same variances. The F ratio should be 1 in this case.
Assumptions of ANOVA. 1) Observations within each sample are independent. 2) Populations from which samples are obtained are normal . 3) Homogeneity of variance. Populations from which samples are obtained have the same variance.
Between treatments variance. The differences between sample means of two populations.
Within treatments variance. The differences within one population.
Within treatments variance is caused by random, non-systematic factors.
Between treatments can be explained by sampling error. It can also be caused by true differences in treatments, called treatment effects.
Total sum of squares = SStotal = ∑X-squared - (∑X)-squared/N
Total degrees of freedom. N - 1.
Degrees of freedom within. n - k. k is the number of treatments.
Degrees of freedom between treatments. k - 1.
Mean square is used in place of variance in ANOVA. It is referred to as MS.
ANOVA hypothesis testing.
Step 1. State hyphothesis and alpha level. Ex. µ1 = µ2 =µ3.
Step 2. Locate the critical region. Ex. df within is 15. df between is 2. The F ratio for df 2, 15 and .05 alpha is 3.68.
Step 3. Calculate the F ratio. Ex. 7.16 in above example.
Step 4. Make a decision. 7.16 is higher than 3.68, so we reject the null hypothesis and conclude there are significant differences among population means.
To calculate effect size, ANOVA uses eta-squared versus r-squared.
Post-hoc tests for ANOVA. Determines which mean differences are significant and which ones are not. Used when evaluating three or more treatments,
Tukey's HSD test. Assumes treatment conditions are honestly significantly different (HSD).
Shuffe test. Used to compare only the two mean differences that you wish to compare in a study. Calculates an F-ratio similar to that of Tukey's.
F = t-squared. F measures variances, while t measures distances.