Logistic regression - overview
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Logistic regression | Spearman's rho | Cochran's Q test |
You cannot compare more than 3 methods |
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Independent variables | Variable 1 | 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 within subject factor ($\geq 2$ related groups) | |
Dependent variable | Variable 2 | Dependent variable | |
One categorical with 2 independent groups | One of ordinal level | One categorical with 2 independent groups | |
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: $\pi_1 = \pi_2 = \ldots = \pi_I$
Here $\pi_1$ is the population proportion of 'successes' for group 1, $\pi_2$ is the population proportion of 'successes' for group 2, and $\pi_I$ is the population proportion of 'successes' for group $I.$ | |
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: not all population proportions are equal | |
Assumptions | Assumptions | Assumptions | |
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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. | If a failure is scored as 0 and a success is scored as 1:
$Q = k(k - 1) \dfrac{\sum_{groups} \Big (\mbox{group total} - \frac{\mbox{grand total}}{k} \Big)^2}{\sum_{blocks} \mbox{block total} \times (k - \mbox{block total})}$ Here $k$ is the number of related groups (usually the number of repeated measurements), a group total is the sum of the scores in a group, a block total is the sum of the scores in a block (usually a subject), and the grand total is the sum of all the scores. Before computing $Q$, first exclude blocks with equal scores in all $k$ groups. | |
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 $Q$ 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 | If the number of blocks (usually the number of subjects) is large, approximately the chi-squared distribution with $k - 1$ degrees of freedom | |
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:
| If the number of blocks is large, the table with critical $X^2$ values can be used. If we denote $X^2 = Q$:
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Wald-type approximate $C\%$ confidence interval for $\beta_k$ | n.a. | 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). | - | - | |
Goodness of fit measure $R^2_L$ | 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 | |
- | - | Friedman test, with a categorical dependent variable consisting of two independent groups. | |
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? | Subjects perform three different tasks, which they can either perform correctly or incorrectly. Is there a difference in task performance between the three different tasks? | |
SPSS | SPSS | SPSS | |
Analyze > Regression > Binary Logistic...
| Analyze > Correlate > Bivariate...
| Analyze > Nonparametric Tests > Legacy Dialogs > K Related Samples...
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Jamovi | Jamovi | Jamovi | |
Regression > 2 Outcomes - Binomial
| Regression > Correlation Matrix
| Jamovi does not have a specific option for the Cochran's Q test. However, you can do the Friedman test instead. The $p$ value resulting from this Friedman test is equivalent to the $p$ value that would have resulted from the Cochran's Q test. Go to:
ANOVA > Repeated Measures ANOVA - Friedman
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Practice questions | Practice questions | Practice questions | |