F test 和 t test 小总结

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**?

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Pexels / Pixabay[/caption]

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.

KaTeX parse error: Expected 'EOF', got ' ' at position 9: 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.

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] })

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

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;

Except for F=t2F=t^2F=t2, I summarized a table for F-test and t-test.

##t- test & F-test Assumption ##

1. Observations are Independent and Random
2. The population are Normally distributed
3. No outliers

####t-test Null-hypothesis:
The mean value of the two groups are the same.
The mean value = n0.

The mean value of three or more groups are the same.
N1=N2=N3…

The Standard deviation is not known and Sample size is small.
F-test Features:
The variance of the normal populations is not known.

1.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.