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How to do a hypothesis test of a coefficient’s significance (in R)

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Let’s say we have a linear model, either one variable or many. How do we conduct a test of significance for the coefficient of a single explanatory variable in the model? Similarly, how can we determine if an explanatory variable has a significant impact on the response variable?

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We will use the fake data shown below with a single variable model. You can use a model created from your own actual data instead.

x <- c( 34,   9,  78,  60,  22,  45,  83,  59,  25)
y <- c(126, 347, 298, 309, 450, 187, 266, 385, 400)
model <- lm(y ~ x)

We can test whether a coefficient is zero by using that as our null hypothesis, $H_0: \beta_i = 0$. We can use any value $0 \le \alpha \le 1$ as our Type 1 error rate; we will set $\alpha$ to be 0.05 here.

The answer to our hypothesis test can be obtained by looking at just the coefficients portion of the model summary:

            Estimate   Std. Error t value    Pr(>|t|)   
(Intercept) 354.082248 76.732772   4.6144853 0.002441995
x            -1.009013  1.472939  -0.6850334 0.515358250

The final column of output shows $p$-values for each $\beta_i$. The $p$-value associated with the $x$ row is therefore for $\beta_1$, the coefficient on $x$. Because it is 0.515358250, which is greater than $\alpha$, we cannot reject the null hypothesis, and we should continue to assume that $\beta_1=0$ and there is no significant relationship between the explanatory and response variable in this situation.

Content last modified on 24 July 2023.

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Contributed by Elizabeth Czarniak (