# How to do a hypothesis test of a coefficient’s significance

## Description

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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## Solution, in R

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.

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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:

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summary(model)$coef

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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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