**Introduction**

So far statistical inference was confined to input variable that could take up two possible values (two sample tests), or there was no notion of an input variable (single sample tests).

ANOVA

- When there are three or more states of a single variable we can use ANOVA

Chi-Square Test of Independence

- Can be used when we want to compare multiple propotions

**Basic of ANOVA**

Tests of hypothesis that: Î¼A=Î¼B=Î¼C=Î¼D

Take the table:

**ANOVA Output**

Compare F calculated against the F-distribution with a-1, N-a degrees of freedom and get a p-value.

**Why F for difference in means??**

- The F is the ratio of two variances (where the sample come from a normal distribution and the null hypothesis is that the variances are equal)
- MSB is a way of calculating total variance
- MSE is a way of calculating total variance
- MSB MSE and MST will be equal if the null hypothesis is true.
- However is the null hypothesis is not, then MSB > MST > MSE.

**MSB and MSE**

**Why do you do after rejecting the null hypothesis**

Need to figure out which pairs of treatments are different. One popular way is the Tukey test.

Method 1:

- Decide on an alpha value

- Calculate the critical tukey distance with this formula:

- Do a complete enumeration of all pairs of difference in means. i.e., All possible ABS (yi.....yj).

- See which of these are above the critical distance.

**Chi-Square TOI**

When using categorical variables

Use this to test:

- Does the input categorical variable effect the output categorical variable (works 2 or more states of the input or output variable)
- Independence between two variable.
- Construct a contingency table:

- Create the value for this table in accordance to the assumption of independence
- It can be done row wise or column wise, but each cell gets an expected value
- Then if the null hypothesis is true then the test statistic is:

With (r-1) *(c-1) degrees of freedom (or rc-c-r +1)

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