Reading a Multiple Regression Coefficient Table

Introduction

Software can fit a multiple regression model and display a compact table of results in seconds. Reading that table well takes more care: an estimate has meaning only with its units and comparison held fixed, and a small p-value is evidence about one coefficient under a specified model, not a verdict on the model as a whole.

This tutorial explains the usual coefficient-table columns: the estimated coefficients, their standard errors, t statistics, p-values, and confidence intervals. The examples use a model with quantitative predictors and a two-category predictor, so the interpretations include both changes per unit and differences from a reference category.

Model and Assumptions Behind the Table

Suppose a hypothetical sample contains 50 students. The outcome \(Y\) is final-exam score in points. The predictors are age in years, weekly study hours, a baseline exam score in points, and program participation. Program is coded 1 for “yes” and 0 for “no,” making “no” the reference category. A fitted model is

Y_i=\beta_0+\beta_1\text{Age}_i+\beta_2\text{Program}_i+ \beta_3\text{StudyHours}_i+\beta_4\text{Baseline}_i+\varepsilon_i.

Here \(\beta_0\) is the intercept, and each \(\beta_j\) for \(j=1,\ldots,4\) is a population regression coefficient. The model describes the conditional mean of the outcome: the expected score for students with the specified predictor values. Because it has five fitted coefficients and 50 observations, its residual degrees of freedom are \(df=50-5=45\).

The usual t tests and confidence intervals in this table rely on a correctly specified linear conditional-mean model, independent errors with mean zero and constant variance, and normally distributed errors for exact small-sample t inference. The predictor coding and included terms must match the intended model. In practice, the sampling or study design must also support the intended population interpretation; a regression association alone does not establish causation. The table itself cannot verify these conditions.

Definition: In a multiple regression coefficient table, an estimate is the fitted value of a model coefficient; its standard error estimates the sampling variability of that coefficient estimate. A coefficient’s t statistic is its estimate minus the null value, divided by its standard error. The p-value measures how unusual a t statistic at least as extreme would be if the null hypothesis and model assumptions held. A coefficient confidence interval gives a range of values for the population coefficient compatible with the data and model at the stated confidence level.

A Coefficient Table: What Each Column Says

The following hypothetical software output reports two-sided tests of \(H_0:\beta_j=0\) and 95% confidence intervals. Values are rounded; small differences can occur because software performs calculations at greater precision than the displayed estimates and standard errors.

Term	Estimate	SE	t	p-value	95% confidence interval
Intercept	12.40	2.10	5.905	<0.0001	(8.170, 16.630)
Age (years)	0.180	0.060	3.000	0.00439	(0.0592, 0.3008)
Program (yes vs no)	-1.20	0.55	-2.182	0.03439	(-2.308, -0.0922)
Study hours (hours/week)	0.420	0.180	2.333	0.0242	(0.0575, 0.7825)
Baseline score (points)	0.650	0.090	7.222	<0.0001	(0.4687, 0.8313)

The estimate column is in the units of the outcome per unit of the predictor, except for the intercept, which is in outcome units. The SE column is in the same units as its coefficient estimate. For example, the age coefficient and its SE are measured in final-exam points per year. The t statistic is unitless, and the p-value is a probability, so neither has outcome units.

Worked Example: Interpreting Quantitative Coefficients

Use the age and study-hours rows to describe the fitted relationships, making clear what is held fixed.

The estimated age coefficient is \(0.180\). In this model, students differing by one year of age but having the same program status, weekly study hours, and baseline score are estimated to differ by \(0.180\) final-exam points on average, with the older student’s predicted score higher. The units are final-exam points per year. This is an adjusted association, not a claim that becoming a year older causes a score increase.

The estimated study-hours coefficient is \(0.420\). For students with the same age, program status, and baseline score, a one-hour-per-week difference in study time corresponds to an estimated \(0.420\)-point difference in mean final-exam score. Again, the coefficient describes the fitted model’s conditional mean, not necessarily a causal effect.

Worked Example: Interpreting a Categorical Coefficient

Interpret the program estimate of \(-1.20\), given that program participation is coded 1 for yes and 0 for no.

The program coefficient compares “yes” with the reference category “no.” Among students with the same age, study hours, and baseline score, those coded “yes” have a fitted mean final-exam score \(1.20\) points lower than those coded “no.” The sign is negative because the yes group’s adjusted fitted mean is lower in this model.

The coefficient does not compare program participants with everyone in the sample without adjustment, and it does not show that participation lowered scores. The data and design would need to justify a causal interpretation; the coefficient table alone does not do so.

How the t Statistic and p-Value Are Obtained

For an individual coefficient, the usual two-sided test is \(H_0:\beta_j=0\) against \(H_a:\beta_j\ne 0\), conditional on the other predictors in the model. The t statistic compares the estimate with the null value in estimated standard-error units. Under the null hypothesis and the stated regression assumptions, it follows a t distribution with the model’s residual degrees of freedom.

t=\frac{\text{estimate}-\text{null value}}{SE}, \qquad p\text{-value}=2P\left(T_{df}\geq |t|\right),

where \(T_{df}\) denotes a random variable with a t distribution having \(df\) degrees of freedom. The factor of 2 applies to this two-sided test. A one-sided test would require a directional alternative specified before looking at the results, and its p-value would be calculated differently.

Worked Example: Testing the Age Coefficient

Test whether age has a nonzero partial regression coefficient in the displayed model at \(\alpha=0.05\).

The hypotheses are \(H_0:\beta_{\text{Age}}=0\) and \(H_a:\beta_{\text{Age}}\ne0\), with the other included predictors held in the model. The table gives an estimate of \(0.180\), an SE of \(0.060\), and \(df=45\). Thus

t=\frac{0.180-0}{0.060}=3.000.

For a two-sided t test with 45 degrees of freedom, the p-value is \(2P(T_{45}\geq 3.000)\approx0.00439\). Since \(0.00439<0.05\), reject \(H_0\) at the 5% significance level. Under the model assumptions, the data provide evidence that the age coefficient is nonzero after accounting for the other predictors. The p-value does not give the probability that \(H_0\) is true.

Confidence Intervals for Individual Coefficients

A two-sided \(100(1-\alpha)\%\) confidence interval for a coefficient has the form estimate \(\pm\) a t critical value times its SE. For a 95% interval here, the critical value is approximately \(t^*_{45}=2.0141\). The interval estimates the population coefficient, not an individual student’s score.

\text{estimate}\ \pm\ t^*_{df}SE.

The interval is produced by a procedure that, under the model assumptions, covers the fixed population coefficient in 95% of repeated samples. Once a particular interval is computed, it is not correct to say there is a 95% probability that the fixed coefficient lies in that interval.

Worked Example: Interpreting the Program Confidence Interval

Calculate and interpret the 95% confidence interval for the program coefficient.

Using the displayed estimate, its standard error, and \(t^*_{45}\approx2.0141\),

-1.20\pm2.0141(0.55) =-1.20\pm1.1078 =(-2.3078,-0.0922).

Rounded to the table’s precision, the interval is \((-2.308,-0.0922)\) points. In the population represented by the model, the program-yes versus program-no adjusted difference in mean final-exam score is compatible with values from about \(2.308\) points lower to \(0.0922\) points lower, given the model assumptions. Because the entire interval is below zero, it agrees with the two-sided test’s evidence against a zero program coefficient at the 5% level. It does not establish a causal program effect.

A Short Justification: Why the Test and Interval Agree

The confidence interval and two-sided test are two presentations of the same t-based inference. To see the connection for a 95% interval, suppose a proposed coefficient value \(\beta_0\) is tested. The corresponding null is not rejected at the 5% level exactly when

\left|\frac{\text{estimate}-\beta_0}{SE}\right| \leq t^*_{df}.

Multiplying both sides by the positive \(SE\), then rearranging, gives

-t^*_{df}SE \leq \text{estimate}-\beta_0 \leq t^*_{df}SE \quad\Longleftrightarrow\quad \text{estimate}-t^*_{df}SE \leq \beta_0 \leq \text{estimate}+t^*_{df}SE.

The final range is the 95% confidence interval. Thus, zero is excluded from a 95% interval precisely when the corresponding two-sided test of a zero coefficient rejects at \(\alpha=0.05\), apart from small discrepancies caused by rounding.

Reading the Table Carefully

Use the table to answer a specific question rather than treating a column as a complete conclusion. For an individual coefficient, a reliable sequence is to identify the term and its coding, interpret the estimate in context, check the t statistic and p-value for the stated test, and read the interval in the same units. Then state the conclusion conditionally on the model assumptions.

For the program coefficient, for example, the estimate gives the direction and size of the adjusted difference, while the p-value addresses evidence against a zero difference. The interval gives a range of coefficient values compatible with the data under the model. Neither the p-value nor the interval answers whether the model’s assumptions hold, whether important predictors were omitted, or whether a causal conclusion is justified. Those are separate questions.

Common mistake: “The program p-value is below 0.05, so there is a 95% chance the program caused a score decrease.” This is incorrect. The p-value is calculated assuming a zero coefficient and the model assumptions; it is not a probability that the null is true, and statistical significance does not establish causation. The appropriate conclusion is that the data provide evidence against a zero adjusted program coefficient under the model, while the causal meaning depends on the study design.

Check Your Understanding

Use the model and table above to answer the following questions.

What does the age estimate of 0.180 mean, and which predictors are held fixed in its interpretation?
Why is the program coefficient a comparison with “no,” and what would change if the reference category were recoded?
For the study-hours row, what does the SE measure, and what units does it have?
In the age test, what does the p-value of 0.00439 quantify? What does it not quantify?
Explain why the program 95% confidence interval and the two-sided test at \(\alpha=0.05\) lead to consistent conclusions.