Stat 3130 statistical methods i
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STAT 3130 Statistical Methods I. Session 4 Two Way Analysis of Variance (ANOVA). Two Way ANOVA. From previous notes, we understand the following about ANOVA:

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STAT 3130 Statistical Methods I

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Stat 3130 statistical methods i

STAT 3130Statistical Methods I

Session 4

Two Way Analysis of Variance (ANOVA)


Two way anova

Two Way ANOVA

From previous notes, we understand the following about ANOVA:

  • It allows us to determine if significant differences of a quantitative variable exist across 3 or more levels of a qualitative variable.

  • What the hypothesis statements look like.

  • The assumptions which need to be met.

  • How to read the ANOVA table.

  • The test statistic and the F-distribution.

  • How to make a decision with the results.

  • If the results allow us to reject the null, we know to run a post hoc test and how to find the differences.


Two way anova1

Two Way ANOVA

But the notes on ANOVA so far, only allow us to evaluate a single qualitative variable (with 3+ levels) with a quantitative variable.

What if we have a two qualitative variables and a single quantitative variable?

For example, what if we wanted to examine how music affects the productivity of our employees? Specifically, we want to examine the type of music (rock, country, jazz) as well as the loudness of the music (soft or loud).

To do this, we would need to run a two way ANOVA.


Two way anova2

Two Way ANOVA

Two Way ANOVA will provide us with not only the main effects (the results of each qualitative variable individually) but it will also provide us with the interaction effects BETWEEN the qualitative variables.

Interaction effects are commonly present, so you need to look for them. Here are some examples:

  • Diet Plan A resulted in a greater weight loss for women than for men, but Diet Plan B resulted in a greater weight loss for men than for women.

  • Fertilizer A was best in high sunlight areas but Fertilizer B was best in low light areas.


Two way anova3

Two Way ANOVA

As we saw with the One Way ANOVA, we have some important assumptions which need to be checked prior to executing a Two Way ANOVA:

  • The samples must have been randomly drawn and must be independent of each other.

  • The variances of the samples must be approximately equal.

  • The groups should all have approximately the same sample size.


Two way anova4

Two Way ANOVA

In a Two Way ANOVA, we actually test three sets of hypothesis statements simultaneously:

H1a: The population means of the first factor are not equal.

H1b: The population means of the second factor are not equal.

H1c: There is an interaction effect between the first and second factors.


Two way anova5

Two Way ANOVA

In a Two Way ANOVA we are working to ascertain the following:

  • What is the variability in the data attributable to Factor A?

  • What is the variability in the data attributable to Factor B?

  • What is the variability in the data attributable to the interaction between Factors A and B?

  • What is the variability in the data which is random and cannot be attributed to A, B or A and B?


Two way anova6

Two Way ANOVA

There are three possible outcomes from your analysis…

1. Significant main effects but no significant interaction

2. One significant main effect, one nonsignificant main effect, and significant interaction

3. Significant main effects and significant interaction

Lets look at these outcomes using SAS…


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