5.5 Inference about model parameters
Once the model fit has been evaluated and the fulfillment of the assumptions associated with the errors has been checked, the next step is to determine the statistical significance of the estimated parameters. This analysis makes it possible to establish whether the covariates included in the model provide relevant information for explaining the variability of the response variable.
In models fitted with data from complex surveys, hypothesis tests for regression coefficients are constructed using variance estimators that are consistent with the sampling design. To evaluate the effect of a covariate associated with the parameter \(\beta_j\), the following contrast is usually considered
\[ H_0:\beta_j=0 \qquad \text{frente a} \qquad H_1:\beta_j\neq 0. \]
The corresponding test statistic is defined as
\[ t_j= \frac{\hat{\beta}_{j}-\beta_{j}} {\sqrt{\widehat{Var}(\hat{\beta}_{j})}} \]
which, under the null hypothesis, approximately follows a Student’s \(t\) distribution with \(df\) degrees of freedom associated with the sampling design (usually defined as the number of primary sampling units minus the number of strata). The inferential decision is based on comparing the absolute value of the statistic \(t_j\) with the critical value of the reference distribution. When the observed value of the statistic is sufficiently large in magnitude, the null hypothesis is rejected and it is concluded that the corresponding covariate has a statistically significant association with the response variable.
The same distributional properties make it possible to construct confidence intervals for the regression coefficients. Thus, a confidence interval at the \((1-\alpha)\times100\%\) level for the parameter \(\beta_j\) is given by
\[ \hat{\beta}_{j} \pm t_{1-\frac{\alpha}{2},\,df} \sqrt{\widehat{Var}(\hat{\beta}_{j})}, \]
where \(t_{1-\frac{\alpha}{2},\,df}\) represents the quantile of order \(1-\alpha/2\) of the Student’s \(t\) distribution with \(df\) degrees of freedom.
To implement these procedures in R, the functions summary.svyglm and confint.svyglm are used; these make it possible to obtain, respectively, significance tests based on the \(t\) statistic and confidence intervals for the coefficients estimated under the complex sampling design. The corresponding results are presented in Table 5.3.
summary_table <- summary(fit_svy)$coefficients %>%
as.data.frame() %>%
tibble::rownames_to_column("Parameter")| Parameter | Estimate | Std. Error | t value | Pr(>|t|) |
|---|---|---|---|---|
| (Intercept) | 73.580 | 59.470 | 1.237 | 0.218 |
| Expenditure | 1.222 | 0.197 | 6.212 | 0.000 |
| ZoneUrban | 66.652 | 39.666 | 1.680 | 0.096 |
| SexMale | 20.644 | 15.611 | 1.322 | 0.189 |
The model results indicate that the variable Expenditure has a statistically significant association with the variable of interest (Income). This effect is highly statistically significant (\(p < 0.001\)), providing strong evidence of a positive linear relationship between the two variables. In turn, for the categorical variables ZoneUrban and SexMale, the level of statistical significance is marginal (\(p = 0.096\) and \(p = 0.189\), respectively), so the evidence is not conclusive at the conventional 5% level.