# R lm output - t values and Pr(>|t|) don't correspond

I have just run a lm in R and here is the output:

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -1.182e+01  1.558e+01  -0.759   0.4482
ht           1.978e-01  1.909e-01   1.036   0.3003
educ         5.893e-02  7.564e-03   7.791 1.37e-14 ***
age          4.022e-03  2.361e-03   1.704   0.0887 .
Hb           5.329e-02  1.845e-01   0.289   0.7728
I(Hb^2)      2.048e-04  6.624e-03   0.031   0.9753
I(ht^2)     -5.409e-04  5.883e-04  -0.919   0.3581


Can someone please explain to me why for this line:

I(Hb^2)      2.048e-04  6.624e-03   0.031   0.9753


I have a t value of 0.031 and this corresponds to a P value of 0.97? A t value this small is close to 0 meaning that I would expect for it to have a probability P('less than' t) close to 50%, maybe 52%? I have made an awful drawing of my understanding here: I think my flawed understanding has something to do with the fact that R's output gives P(>|t|)?

• Perhaps it might help to edit your question to say what you think the $p$-value should be especially if you do not understand @peterflom's answer. – mdewey Oct 21 '18 at 16:12
• good suggestion, thank you and I have made this edit. – tbrick Oct 21 '18 at 16:14
• Hint: graph the interval (on the horizontal axis) corresponding to points whose sizes (absolute values) exceed 0.031. As a check, obviously this interval will not include 0, which rules out your graphic. – whuber Oct 21 '18 at 20:31

One thing you might be missing is that R performs a two-tailed t-test - the probability of a test statistic whose value is more extreme than the observed value, i.e. whose absolute value is $$\geq$$ the (absolute value of the) observed value.

The open space in the middle of this curve is the region with $$-0.031 < t < 0.031$$; the pink and blue areas, which add up to 0.9753, is the region with $$|t|>0.031$$. png("tcurve.png")
tx <- seq(from=0.031,to=3.5,length=51)
par(yaxs="i")
curve(dt(x,df=10),from=-3.5,to=3.5,axes=FALSE,ylab="")
axis(side=1)
box(bty="l")
tvals <- dt(tx,df=10)
polygon(c(tx,rev(tx)),c(tvals,rep(0,length(tx))),col="pink")
polygon(c(rev(-tx),-tx),c(rev(tvals),rep(0,length(tx))),col="lightblue")
dev.off()


That line makes perfect sense. A t value close to 0 corresponds to a high (that is very nonsignificant) p value.

• Thanks for your answer, please see the edits I have made to my question for what I expected the p value should be. – tbrick Oct 21 '18 at 16:14
• Sorry, but your edit doesn't help. PR (>T) describes what a p value is. What the 0.97 says is that, if the null hypothesis is true, it is still very likely to get a p value this far or farther from 0. When T = 0.00, the p would be 1.00 (approx). When t is far from 0, p will be small. – Peter Flom - Reinstate Monica Oct 21 '18 at 16:23
• Sorry, my edit didnt make any sense because stackexchange hid all my text behind my < sign. Not sure why it did this and I have replaced < with 'less than'. I've also added a terrible drawing for why I think the P value should be ~52%. I didnt actually run this t test but thought it should be something a little larger than 50%. – tbrick Oct 21 '18 at 18:29
• Nope, close to 1. That's because it is two sided, as is indicated by the absolute value sign around the t. – Peter Flom - Reinstate Monica Oct 22 '18 at 0:06