# F-statistics and coefficient p-value of model with only one variable

F-test tests the null hypothesis that all coefficient of variables in the model equal to zero.

P-value in a hypothesis test shows the probability of having observed results if null hypothesis is true.

My question is that if the model only has one variable, just like m2 and m3 shown as following, is the result of f-statistics just the same as the p-value?

Dataset Regarding the dataset. The original dataset has 50 states. Alaska and Texas are the outliers. From some analysis, I already decide to remove Alaska as it has a large effect on the linear model according to the regression results. Now the problem is whether Texas should be removed.

area.no.alaska has 49 states, Alaska is removed area.no.tx.al has 48 states, Alaska and Texas is removed

I am trying to compare m2 with m4 to figure out which is better.

So far, I think m2 is better because it has higher r-square, higher p-value of variable land.area even though both models have p-value very close to zero for variable land.area. I am not sure whether it is necessary to compare f-statistics as only one variable in the model and I already compared the p-value of that variable.

m2 = lm(area.no.alaska$farm.area ~ area.no.alaska$land.area)
m4 = lm(farm.area ~ land.area, data = area.no.al.tx)


summary(m4)

Call:
lm(formula = farm.area ~ land.area, data = area.no.al.tx)

Residuals:
Min     1Q Median     3Q    Max
-41309  -8736  -1784   4361  36054

Coefficients:
Estimate Std. Error t value      Pr(>|t|)
(Intercept) -569.41436 4310.67726  -0.132         0.895
land.area      0.46526    0.06404   7.266 0.00000000365 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 16300 on 46 degrees of freedom
Multiple R-squared:  0.5344,    Adjusted R-squared:  0.5242
F-statistic: 52.79 on 1 and 46 DF,  p-value: 0.000000003646



summary(m2)

Call:
lm(formula = area.no.alaska$farm.area ~ area.no.alaska$land.area)

Residuals:
Min     1Q Median     3Q    Max
-50695  -8712   -228   7825  49354

Coefficients:
Estimate  Std. Error t value Pr(>|t|)
(Intercept)              -8365.76964  4367.14590  -1.916   0.0615 .
area.no.alaska\$land.area     0.62171     0.05716  10.877 1.98e-14 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 18620 on 47 degrees of freedom
Multiple R-squared:  0.7157,    Adjusted R-squared:  0.7096
F-statistic: 118.3 on 1 and 47 DF,  p-value: 1.981e-14


• m4 uses the data "area.no.al.tx" and m2 uses the data "area.no.alaska". How are these data sets different? It is no possible to answer this question without knowing that. Otherwise, the models appear to be the same, so I am not sure why you are getting different t-values for land.area. May 31, 2017 at 14:09
• thanks. I just updated question with description of the datasets May 31, 2017 at 17:55

• I remove Texas because it is an outlier. I build m2 and m4 to decide how to deal with this outlier. Alaska is another outlier. I built a model with Alaska removed, and the regression result shows it is better to removeAlaska. May 31, 2017 at 22:43