How to Easily Perform a t-Test for Regression Slope in R

Theoretical Foundation of the Regression Slope Test

Whenever we perform simple Linear Regression, we are attempting to model the relationship between a single independent variable (X) and a dependent variable (Y). This analysis yields an estimated sample regression equation, which mathematically describes the best-fit line through the observed data points. This line is expressed as:

ŷ = b₀ + b₁x

In this equation, ŷ represents the predicted value of the dependent variable, b₀ is the y-intercept (the predicted value of Y when X is zero), and b₁ is the Slope Coefficient. The slope, b₁, is the critical parameter, as it quantifies the average change in Y associated with a one-unit increase in X. If the true slope in the population (often denoted as β₁) is zero, then X has no linear relationship with Y, meaning the predictor variable holds no predictive power in the model.

Our primary statistical goal is to determine if the calculated sample slope, b₁, provides sufficient evidence to conclude that the true population slope (β₁) is non-zero. This determination requires formal Hypothesis Testing, which assesses the likelihood of observing our calculated slope if, in reality, no underlying relationship existed in the broader population.

Formulating the Null and Alternative Hypotheses

The t-test for the Slope Coefficient is structured around two competing statements concerning the population parameter β₁—the true, underlying slope of the relationship.

The Null Hypothesis (H₀)

The null hypothesis always asserts a lack of effect or relationship. In the context of linear regression, H₀ states that the true slope is zero. If we fail to reject H₀, we conclude that the observed sample relationship is likely a result of random sampling variation or chance.

• H₀: β₁ = 0 (The population slope is zero; the predictor variable has no statistically significant linear effect on the response variable.)

The Alternative Hypothesis (H₁ or Hₐ)

The alternative hypothesis asserts that a statistically significant relationship exists. This means the true slope is not zero. While directional tests (one-tailed) exist, the standard procedure performed by R’s lm() function is a two-tailed test, checking for significance in either the positive or negative direction.

• H₁: β₁ ≠ 0 (The population slope is not zero; a statistically significant linear relationship exists.)

The decision to reject H₀ is fundamentally driven by the probability (P-value) of obtaining our sample data if the null hypothesis were true. If this probability is very low (typically less than 5%), we reject the null hypothesis in favor of the alternative.

Deriving the T-Statistic and Understanding Standard Error

To quantify how far our estimated sample slope b₁ lies from the null value of zero, we calculate the t-statistic. The t-statistic measures this deviation in units of Standard Error. The general formula for the t-statistic of the slope is:

t = (b₁ – β₁_hypothesized) / se(b₁)

Since we test against the null hypothesis that β₁ is zero, the formula simplifies significantly:

t = b₁ / se(b₁)

Where:

• b₁ represents the estimated Slope Coefficient (the value in the “Estimate” column of the R output).
• se(b₁) represents the Standard Error of the slope estimate (the value in the “Std. Error” column).

The Standard Error (se(b₁)) is a critical measure of precision. It estimates the variability of the sample slope estimate across hypothetical repetitions of the sampling process. A small standard error suggests that our estimate b₁ is highly reliable and precise. A large t-statistic (far from zero) indicates that the observed slope is many standard errors away from zero, making it statistically unlikely to have occurred if H₀ were true. This test statistic follows a t-distribution with n-2 degrees of freedom for simple linear regression.

Practical Implementation: Linear Regression in R

R automates the complex calculations involved in the t-test for the slope through the lm() function. We do not need to manually calculate the t-statistic or the P-value; we only need to invoke the model summary.

We will use the following example data frame, containing the hours studied and the final exam scores for 12 students:

Step 1: Data Setup

The dataset includes 12 observations (n=12), establishing the degrees of freedom for the t-test as 12 – 2 = 10. We hypothesize that ‘hours’ predicts ‘score’.

#create data frame
df <- data.frame(hours=c(1, 1, 2, 2, 3, 4, 5, 5, 5, 6, 6, 8),
                 score=c(65, 67, 78, 75, 73, 84, 80, 76, 89, 91, 83, 82))

#view data frame
df

   hours score
1      1    65
2      1    67
3      2    78
4      2    75
5      3    73
6      4    84
7      5    80
8      5    76
9      5    89
10     6    91
11     6    83
12     8    82

Step 2: Fitting the Model and Generating Summary

We use the lm() function to fit the model and then apply summary() to obtain the full regression output, which contains the results of the t-test for both the intercept and the slope:

#fit simple linear regression model
fit <- lm(score ~ hours, data=df)

#view model summary
summary(fit)

Call:
lm(formula = score ~ hours, data = df)

Residuals:
   Min     1Q Median     3Q    Max 
-7.398 -3.926 -1.139  4.972  7.713 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  67.7685     3.3757  20.075 2.07e-09 ***
hours         2.7037     0.7456   3.626  0.00464 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 5.479 on 10 degrees of freedom
Multiple R-squared:  0.568,	Adjusted R-squared:  0.5248 
F-statistic: 13.15 on 1 and 10 DF,  p-value: 0.004641

Interpreting the R Output for Slope Significance

The interpretation centers on the “Coefficients” table within the summary(fit) output. This table provides all the necessary statistics to test the null hypothesis for the Slope Coefficient (‘hours’).

Focusing on the Slope Coefficient (‘hours’)

The estimated regression equation is: Estimated Score = 67.7685 + 2.7037(Hours Studied). We focus on the row corresponding to the ‘hours’ variable:

1. Estimate (b₁): 2.7037. This is the calculated slope, indicating a positive relationship.
2. Std. Error (se(b₁)): 0.7456. This is the measure of the precision of our slope estimate.
3. t value (t): 3.626. This is the calculated t-statistic. It confirms that the observed slope (2.7037) is approximately 3.626 standard errors away from zero.
4. Pr(>|t|) (P-value): 0.00464. This is the probability of observing a t-statistic as extreme as 3.626 (or more extreme) if the null hypothesis (β₁=0) were true.

Making the Statistical Decision

To conclude the t-test, we compare the calculated P-value to the chosen significance level (α), which is almost universally set at 0.05. The decision rule is simple: If P-value < α, reject H₀.

• P-value = 0.00464
• Significance Level (α) = 0.05

Since 0.00464 is substantially less than 0.05, we reject the null hypothesis (H₀: β₁ = 0). We conclude with strong evidence that the Slope Coefficient is statistically significant, meaning that studying hours genuinely influences the final exam score.

The Critical Role of Precision and Confidence Intervals

The precision provided by the Standard Error is fundamental to the t-test’s validity. If the standard error were larger—perhaps if the data were highly scattered or the sample size was small—the t-statistic would decrease, leading to a higher P-value. Thus, a robust finding requires not only a large slope estimate but also a relatively small standard error.

Furthermore, the standard error allows us to construct a confidence interval for the true population slope (β₁). If the 95% confidence interval for the slope does not contain the value zero, the coefficient is deemed statistically significant at the α = 0.05 level. This confidence interval calculation provides an equivalent decision pathway to the t-test, offering a range of plausible values for the true relationship strength. For our example, since we rejected H₀, the 95% confidence interval for the hours coefficient (2.7037) will definitely exclude zero, reinforcing the conclusion derived from the Hypothesis Testing.

Conclusion and Final Summary

The execution of a t-test for the slope of a Linear Regression line in R is streamlined by the output generated from the summary(lm()) function. This process is essential for transitioning from a descriptive observation (the sample slope b₁) to an inferential conclusion about the true population relationship (β₁).

By focusing on the t value and, most importantly, the corresponding Pr(>|t|) column in the coefficient summary, analysts can quickly determine if the predictor variable has a genuine, statistically significant linear impact on the response. The successful rejection of the null hypothesis confirms that the estimated slope is not merely a random fluctuation but represents a meaningful effect worthy of interpretation and reporting.