> library(faraway)
> data(savings)
> head(savings)
             sr pop15 pop75     dpi ddpi
Australia 11.43 29.35  2.87 2329.68 2.87
Austria   12.07 23.32  4.41 1507.99 3.93
Belgium   13.17 23.80  4.43 2108.47 3.82
Bolivia    5.75 41.89  1.67  189.13 0.22
Brazil    12.88 42.19  0.83  728.47 4.56
Canada     8.79 31.72  2.85 2982.88 2.43

Let’s say I want to test the significance of pop15 on sr in the above dataset. I can use anova or lm commands in R. Here is what I will do

> fit <- lm(sr ~ pop15 + pop75 + dpi + ddpi, savings)
> fit.sum <- summary(fit)
> fit1 <- lm(sr ~ pop15 + pop75 + dpi + ddpi, savings)
> fit2 <- lm(sr ~ pop75 + dpi + ddpi, savings)
> anova(fit1, fit2)
Analysis of Variance Table


Model 1: sr ~ pop15 + pop75 + dpi + ddpi
Model 2: sr ~ pop75 + dpi + ddpi
  Res.Df     RSS Df Sum of Sq      F   Pr(>F)
1     45  650.71
2     46  797.72 -1   -147.01 10.167 0.002603 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> fit.sum
Call:
lm(formula = sr ~ pop15 + pop75 + dpi + ddpi, data = savings)


Residuals:
    Min      1Q  Median      3Q     Max
-8.2422 -2.6857 -0.2488  2.4280  9.7509


Coefficients:
              Estimate Std. Error t value Pr(>|t|)
(Intercept) 28.5660865  7.3545161   3.884 0.000334 ***
pop15       -0.4611931  0.1446422  -3.189 0.002603 **
pop75       -1.6914977  1.0835989  -1.561 0.125530
dpi         -0.0003369  0.0009311  -0.362 0.719173
ddpi         0.4096949  0.1961971   2.088 0.042471 *
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1


Residual standard error: 3.803 on 45 degrees of freedom
Multiple R-squared: 0.3385,     Adjusted R-squared: 0.2797
F-statistic: 5.756 on 4 and 45 DF,  p-value: 0.0007904

Ideally t value is sqrt of F value. Why is the t value for pop15 , -sqrt(F) value ? Note that p value matches!!

WHY ?