Logistic regression - overview
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Logistic regression | Spearman's rho | $z$ test for the difference between two proportions | Wilcoxon signed-rank test |
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Independent variables | Variable 1 | Independent/grouping variable | Independent 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 2 independent groups | 2 paired groups | |
Dependent variable | Variable 2 | Dependent variable | Dependent variable | |
One categorical with 2 independent groups | One of ordinal level | One categorical with 2 independent groups | One quantitative of interval or ratio level | |
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: $\pi_1 = \pi_2$
Here $\pi_1$ is the population proportion of 'successes' for group 1, and $\pi_2$ is the population proportion of 'successes' for group 2. | H0: $m = 0$
Here $m$ is the population median of the difference scores. A difference score is the difference between the first score of a pair and the second score of a pair. Several different formulations of the null hypothesis can be found in the literature, and we do not agree with all of them. Make sure you (also) learn the one that is given in your text book or by your teacher. | |
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 two sided: $\pi_1 \neq \pi_2$ H1 right sided: $\pi_1 > \pi_2$ H1 left sided: $\pi_1 < \pi_2$ | H1 two sided: $m \neq 0$ H1 right sided: $m > 0$ H1 left sided: $m < 0$ | |
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. | $z = \dfrac{p_1 - p_2}{\sqrt{p(1 - p)\Bigg(\dfrac{1}{n_1} + \dfrac{1}{n_2}\Bigg)}}$
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 $\pi_2 > \pi_1.$ | Two different types of test statistics can be used, but both will result in the same test outcome. We will denote the first option the $W_1$ statistic (also known as the $T$ statistic), and the second option the $W_2$ statistic.
In order to compute each of the test statistics, follow the steps below:
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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 $z$ if H0 were true | Sampling distribution of $W_1$ and of $W_2$ 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 standard normal distribution | Sampling distribution of $W_1$:
If $N_r$ is large, $W_1$ is approximately normally distributed with mean $\mu_{W_1}$ and standard deviation $\sigma_{W_1}$ if the null hypothesis were true. Here $$\mu_{W_1} = \frac{N_r(N_r + 1)}{4}$$ $$\sigma_{W_1} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{24}}$$ Hence, if $N_r$ is large, the standardized test statistic $$z = \frac{W_1 - \mu_{W_1}}{\sigma_{W_1}}$$ follows approximately the standard normal distribution if the null hypothesis were true. Sampling distribution of $W_2$: If $N_r$ is large, $W_2$ is approximately normally distributed with mean $0$ and standard deviation $\sigma_{W_2}$ if the null hypothesis were true. Here $$\sigma_{W_2} = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{6}}$$ Hence, if $N_r$ is large, the standardized test statistic $$z = \frac{W_2}{\sigma_{W_2}}$$ follows approximately the standard normal distribution if the null hypothesis were true. If $N_r$ is small, the exact distribution of $W_1$ or $W_2$ should be used. Note: if ties are present in the data, the formula for the standard deviations $\sigma_{W_1}$ and $\sigma_{W_2}$ is more complicated. | |
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:
| Two sided:
| For large samples, the table for standard normal probabilities can be used: Two sided:
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Wald-type approximate $C\%$ confidence interval for $\beta_k$ | n.a. | Approximate $C\%$ confidence interval for $\pi_1 - \pi_2$ | n.a. | |
$b_k \pm z^* \times SE_{b_k}$ 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):
| - | |
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. | Equivalent to | n.a. | |
- | - | 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 the proportion of smokers different between men and women? Use the normal approximation for the sampling distribution of the test statistic. | Is the median of the differences between the mental health scores before and after an intervention different from 0? | |
SPSS | SPSS | SPSS | SPSS | |
Analyze > Regression > Binary Logistic...
| Analyze > Correlate > Bivariate...
| 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...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
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Jamovi | Jamovi | Jamovi | Jamovi | |
Regression > 2 Outcomes - Binomial
| Regression > Correlation Matrix
| 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
| T-Tests > Paired Samples T-Test
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Practice questions | Practice questions | Practice questions | Practice questions | |