Parameter Estimate Tables

Study guides: Exam MAS-I, Exam PCPA

Parameter Estimate Tables (also called coefficient tables or output tables) are standard output from statistical software when fitting regression or Generalized Linear Models. They summarize the estimated model coefficients, their uncertainty, and hypothesis test results.

Typical columns in a parameter estimate table:

Column	Description
Parameter	Predictor name or level
Estimate ( $\hat{\beta}$ )	Maximum likelihood estimate of the coefficient
Std. Error	Estimated standard deviation of $\hat{\beta}$
z-value or t-value	$\hat{\beta} / \text{SE}(\hat{\beta})$ ; test statistic for $H_0: \beta = 0$
p-value	Significance of the predictor: $P(\lvert T_{n-p-1} \rvert \geq \lvert t\text{-value}\rvert)$ for a t-value, $P(\lvert Z \rvert \geq \lvert z\text{-value}\rvert)$ for a z-value
95% CI	$\hat{\beta} \pm t_{0.025,\, n-p-1} \cdot \text{SE}(\hat{\beta})$ for a t-value, $\hat{\beta} \pm 1.96 \cdot \text{SE}(\hat{\beta})$ for a z-value

Which distribution applies: a linear regression table reports t-values — $\sigma^2$ is estimated from the residuals, so under Normal errors $\hat\beta_j/\text{SE}(\hat\beta_j)$ follows a $t$ distribution with $n-p-1$ degrees of freedom ( $p$ predictors plus an intercept). A GLM table such as Poisson or logistic regression reports z-values: Wald statistics referred to the standard Normal, because the maximum likelihood estimate is approximately Normal in large samples. The two agree as the degrees of freedom grow — the two-sided 5% critical value is $2.09$ at 20 df and $1.98$ at 100 df, against $1.96$ — but with few residual df the $t$ multiplier is much larger ( $3.18$ at 3 df)
A small p-value (e.g., $< 0.05$ ) indicates the predictor is statistically significant
For a log-Link Function model (Poisson or Gamma GLM), $e^{\hat{\beta}}$ is the multiplicative factor on the mean per unit increase in the predictor
The reference level of a categorical predictor is absorbed into the intercept; other levels are interpreted relative to it
Standard errors from GLMs are based on the Fisher information matrix

Interpreting GLM Output A Poisson GLM output shows: Intercept $\hat{\beta}_0 = -1.8$ (SE = 0.2) and Territory B $\hat{\beta}_1 = 0.45$ (SE = 0.15, p = 0.003). The reference is Territory A. Interpret the territory coefficient.

Answer The p-value of 0.003 < 0.05 indicates that Territory B is significantly different from Territory A. With a log link, the expected claim count in Territory B is $e^{0.45} \approx 1.57$ times that in Territory A — i.e., approximately 57% more claims in Territory B, all else equal.