Logistic regression - overview
This page offers structured overviews of one or more selected methods. Add additional methods for comparisons (max. of 3) by clicking on the dropdown button in the right-hand column. To practice with a specific method click the button at the bottom row of the table
Logistic regression | McNemar's test | One sample $z$ test for the mean |
You cannot compare more than 3 methods |
---|---|---|---|
Independent variables | Independent 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 | 2 paired groups | None | |
Dependent variable | Dependent variable | Dependent variable | |
One categorical with 2 independent groups | One categorical with 2 independent groups | One quantitative of interval or ratio level | |
Null hypothesis | Null hypothesis | Null hypothesis | |
Model chi-squared test for the complete regression model:
| Let's say that the scores on the dependent variable are scored 0 and 1. Then for each pair of scores, the data allow four options:
Other formulations of the null hypothesis are:
| H0: $\mu = \mu_0$
Here $\mu$ is the population mean, and $\mu_0$ is the population mean according to the null hypothesis. | |
Alternative hypothesis | Alternative hypothesis | Alternative hypothesis | |
Model chi-squared test for the complete regression model:
| The alternative hypothesis H1 is that for each pair of scores, P(first score of pair is 0 while second score of pair is 1) $\neq$ P(first score of pair is 1 while second score of pair is 0). That is, the probability that a pair of scores switches from 0 to 1 is not the same as the probability that a pair of scores switches from 1 to 0. Other formulations of the alternative hypothesis are:
| H1 two sided: $\mu \neq \mu_0$ H1 right sided: $\mu > \mu_0$ H1 left sided: $\mu < \mu_0$ | |
Assumptions | Assumptions | Assumptions | |
|
|
| |
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 $\beta_k$:
| $X^2 = \dfrac{(b - c)^2}{b + c}$
Here $b$ is the number of pairs in the sample for which the first score is 0 while the second score is 1, and $c$ is the number of pairs in the sample for which the first score is 1 while the second score is 0. | $z = \dfrac{\bar{y} - \mu_0}{\sigma / \sqrt{N}}$
Here $\bar{y}$ is the sample mean, $\mu_0$ is the population mean according to the null hypothesis, $\sigma$ is the population standard deviation, and $N$ is the sample size. The denominator $\sigma / \sqrt{N}$ is the standard deviation of the sampling distribution of $\bar{y}$. The $z$ value indicates how many of these standard deviations $\bar{y}$ is removed from $\mu_0$. | |
Sampling distribution of $X^2$ and of the Wald statistic 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:
| If $b + c$ is large enough (say, > 20), approximately the chi-squared distribution with 1 degree of freedom. If $b + c$ is small, the Binomial($n$, $P$) distribution should be used, with $n = b + c$ and $P = 0.5$. In that case the test statistic becomes equal to $b$. | Standard normal distribution | |
Significant? | Significant? | Significant? | |
For the model chi-squared test for the complete regression model and likelihood ratio chi-squared test for individual $\beta_k$:
| For test statistic $X^2$:
| Two sided:
| |
Wald-type approximate $C\%$ confidence interval for $\beta_k$ | n.a. | $C\%$ confidence interval for $\mu$ | |
$b_k \pm z^* \times SE_{b_k}$ 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). | - | $\bar{y} \pm z^* \times \dfrac{\sigma}{\sqrt{N}}$
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). The confidence interval for $\mu$ can also be used as significance test. | |
Goodness of fit measure $R^2_L$ | n.a. | Effect size | |
$R^2_L = \dfrac{D_{null} - D_K}{D_{null}}$ 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. | - | Cohen's $d$: Standardized difference between the sample mean and $\mu_0$: $$d = \frac{\bar{y} - \mu_0}{\sigma}$$ Cohen's $d$ indicates how many standard deviations $\sigma$ the sample mean $\bar{y}$ is removed from $\mu_0.$ | |
n.a. | n.a. | Visual representation | |
- | - | ||
n.a. | Equivalent to | n.a. | |
- |
| - | |
Example context | Example context | Example context | |
Can body mass index, stress level, and gender predict whether people get diagnosed with diabetes? | Does a tv documentary about spiders change whether people are afraid (yes/no) of spiders? | Is the average mental health score of office workers different from $\mu_0 = 50$? Assume that the standard deviation of the mental health scores in the population is $\sigma = 3.$ | |
SPSS | SPSS | n.a. | |
Analyze > Regression > Binary Logistic...
| Analyze > Nonparametric Tests > Legacy Dialogs > 2 Related Samples...
| - | |
Jamovi | Jamovi | n.a. | |
Regression > 2 Outcomes - Binomial
| Frequencies > Paired Samples - McNemar test
| - | |
Practice questions | Practice questions | Practice questions | |