# F-Test and t-Test

### Data Science Day 21: F -test and t-test

From last time we know t-test is used for comparing the mean of 2-level categorical variable and ANOVA is used for comparing the mean value of a 3-level categorical variable or more.

Question:

However, there is a question bugs me, why both T-test and ANOVA are comparing the mean value, but one P-value comes from the t-test and the other P-value is derived from the F-test?

Pexels / Pixabay

I did a bit research into this and discussed with little Rain, then we found out the key relation to answer is the equivalence of F and t-test.

F= t^{2}

The hidden reason is when pair of the sample are normally distributed then the ratios of variance of sample in each pair will always follow the same distribution. Therefore, the t-test and F-test generate the same p-values.

#### Example : F-test vs t-test in Blood pressure decrease dataset

We want to know if the blood pressure medication has changed the blood pressure for 15 patients after 6 months.

```test=pd.DataFrame({"score_decrease": [ -5, -8, 0, 0, 0 ,2,4,6,8, 10,10, 10,18,26,32] })
center=pd.DataFrame({"score_remained": [ 0, 0, 0, 0, 0 ,0,0,0,0, 0,0, 0,0,0,0] })```

#### F-test results:

```scipy.stats.f_oneway(score_decrease,score_remained)
F_onewayResult(statistic=array([ 7.08657734]), pvalue=array([ 0.01272079]))```

#### t-test results:

```scipy.stats.ttest_ind(score_decrease, score_remained)
Ttest_indResult(statistic=array([ 2.66206261]), pvalue=array([ 0.01272079]))```

As we can see the F-test and t-test have the same P-value= 0.0127.

I used SAS to generate a graph:

```ods graphics on;
proc ttest h0=0 plots(showh0) sides=u alpha=0.05;
var decrease;
run;
ods graphics off;```

#### Summary:

Except F=t^2, I summarized a table for F-test and t-test.

basic comparisont testF test
Assumption1. Observations are Independent and Random
2. Population are Normally distributed
3. No outliers
1. Observations are Independent and Random
2. Population are Normally distributed
3. No outliers
Null-hypothesisThe mean value of two groups are the same.
The mean value = n0.
The mean value of three or more groups are the same.
N1=N2=N3...
Featurestandard deviation is not known. Sample size is smallthe variance of the normal populations is not known
Application1.Compare mean value of two groups.
2.Compare mean value of a group with a particular number.
1. comparing the variances of two or more populations.
2. ANOVA comparing the mean value of 3 or more groups.