Logistic regression - overview
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Logistic regression | Spearman's rho | Chi-squared test for the relationship between two categorical variables | $z$ test for the difference between two proportions |
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Independent variables | Variable 1 | Independent /column variable | Independent/grouping variable | |
One or more quantitative of interval or ratio level and/or one or more categorical with independent groups, transformed into code variables | One of ordinal level | One categorical with $I$ independent groups ($I \geqslant 2$) | One categorical with 2 independent groups | |
Dependent variable | Variable 2 | Dependent /row variable | Dependent variable | |
One categorical with 2 independent groups | One of ordinal level | One categorical with $J$ independent groups ($J \geqslant 2$) | One categorical with 2 independent groups | |
Null hypothesis | Null hypothesis | Null hypothesis | Null hypothesis | |
Model chi-squared test for the complete regression model:
| H0: $\rho_s = 0$
Here $\rho_s$ is the Spearman correlation in the population. The Spearman correlation is a measure for the strength and direction of the monotonic relationship between two variables of at least ordinal measurement level. In words, the null hypothesis would be: H0: there is no monotonic relationship between the two variables in the population. | H0: there is no association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
| H0: $\pi_1 = \pi_2$
Here $\pi_1$ is the population proportion of 'successes' for group 1, and $\pi_2$ is the population proportion of 'successes' for group 2. | |
Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
Model chi-squared test for the complete regression model:
| H1 two sided: $\rho_s \neq 0$ H1 right sided: $\rho_s > 0$ H1 left sided: $\rho_s < 0$ | H1: there is an association between the row and column variable More precisely, if there are $I$ independent random samples of size $n_i$ from each of $I$ populations, defined by the independent variable:
| H1 two sided: $\pi_1 \neq \pi_2$ H1 right sided: $\pi_1 > \pi_2$ H1 left sided: $\pi_1 < \pi_2$ | |
Assumptions | Assumptions | Assumptions | Assumptions | |
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Test statistic | Test statistic | Test statistic | Test statistic | |
Model chi-squared test for the complete regression model:
The wald statistic can be defined in two ways:
Likelihood ratio chi-squared test for individual $\beta_k$:
| $t = \dfrac{r_s \times \sqrt{N - 2}}{\sqrt{1 - r_s^2}} $ Here $r_s$ is the sample Spearman correlation and $N$ is the sample size. The sample Spearman correlation $r_s$ is equal to the Pearson correlation applied to the rank scores. | $X^2 = \sum{\frac{(\mbox{observed cell count} - \mbox{expected cell count})^2}{\mbox{expected cell count}}}$
Here for each cell, the expected cell count = $\dfrac{\mbox{row total} \times \mbox{column total}}{\mbox{total sample size}}$, the observed cell count is the observed sample count in that same cell, and the sum is over all $I \times J$ cells. | $z = \dfrac{p_1 - p_2}{\sqrt{p(1 - p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
Here $p_1$ is the sample proportion of successes in group 1: $\dfrac{X_1}{n_1}$, $p_2$ is the sample proportion of successes in group 2: $\dfrac{X_2}{n_2}$, $p$ is the total proportion of successes in the sample: $\dfrac{X_1 + X_2}{n_1 + n_2}$, $n_1$ is the sample size of group 1, and $n_2$ is the sample size of group 2. Note: we could just as well compute $p_2 - p_1$ in the numerator, but then the left sided alternative becomes $\pi_2 < \pi_1$, and the right sided alternative becomes $\pi_2 > \pi_1.$ | |
Sampling distribution of $X^2$ and of the Wald statistic if H0 were true | Sampling distribution of $t$ if H0 were true | Sampling distribution of $X^2$ if H0 were true | Sampling distribution of $z$ if H0 were true | |
Sampling distribution of $X^2$, as computed in the model chi-squared test for the complete model:
| Approximately the $t$ distribution with $N - 2$ degrees of freedom | Approximately the chi-squared distribution with $(I - 1) \times (J - 1)$ degrees of freedom | Approximately the standard normal distribution | |
Significant? | Significant? | Significant? | Significant? | |
For the model chi-squared test for the complete regression model and likelihood ratio chi-squared test for individual $\beta_k$:
| Two sided:
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| Two sided:
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Wald-type approximate $C\%$ confidence interval for $\beta_k$ | n.a. | n.a. | Approximate $C\%$ confidence interval for $\pi_1 - \pi_2$ | |
$b_k \pm z^* \times SE_{b_k}$ where the critical value $z^*$ is the value under the normal curve with the area $C / 100$ between $-z^*$ and $z^*$ (e.g. $z^*$ = 1.96 for a 95% confidence interval). | - | - | Regular (large sample):
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Goodness of fit measure $R^2_L$ | n.a. | n.a. | n.a. | |
$R^2_L = \dfrac{D_{null} - D_K}{D_{null}}$ There are several other goodness of fit measures in logistic regression. In logistic regression, there is no single agreed upon measure of goodness of fit. | - | - | - | |
n.a. | n.a. | n.a. | Equivalent to | |
- | - | - | When testing two sided: chi-squared test for the relationship between two categorical variables, where both categorical variables have 2 levels. | |
Example context | Example context | Example context | Example context | |
Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes? | Is there a monotonic relationship between physical health and mental health? | Is there an association between economic class and gender? Is the distribution of economic class different between men and women? | Is the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic. | |
SPSS | SPSS | SPSS | SPSS | |
Analyze > Regression > Binary Logistic...
| Analyze > Correlate > Bivariate...
| Analyze > Descriptive Statistics > Crosstabs...
| SPSS does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Analyze > Descriptive Statistics > Crosstabs...
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Jamovi | Jamovi | Jamovi | Jamovi | |
Regression > 2 Outcomes - Binomial
| Regression > Correlation Matrix
| Frequencies > Independent Samples - $\chi^2$ test of association
| Jamovi does not have a specific option for the $z$ test for the difference between two proportions. However, you can do the chi-squared test instead. The $p$ value resulting from this chi-squared test is equivalent to the two sided $p$ value that would have resulted from the $z$ test. Go to:
Frequencies > Independent Samples - $\chi^2$ test of association
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Practice questions | Practice questions | Practice questions | Practice questions | |