ANCOVA allows the user to analyze the difference between multiple group means, while taking into account the effect of variables that have an influence on the dependent variable but are not part of the experimental manipulation (i.e., covariates).
Components and model terms:
Sum of Squares: There are different types of the sum of squares. The choice of the type is important when there are multiple factors and when the data are unbalanced. In an unbalanced design, the different levels of the independent variable do not contain an equal number of observations (e.g., one group contains more obsevations than another group). In this scenario, the sum of squares type can influence the results.
For each independent variable, a specific contrast can be selected by clicking on none
in the right column.
Factors: These are the independent variables included in the analysis (i.e., the variables selected in the Fixed Factors
box).
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bootstraps: By selecting this option, the bootstrapped post hoc test is applied. By default, the number of replications is set to 1000. This can be changed into the desired number. Standard
. Dunnett: When selecting this method, all the levels are compared to one specific level, for example to the control group. At the moment, it is not possible to manually specify to which level the others levels are compared, but it is based on the order of the levels. To change the order of the levels, the level labels can be adjusted.
GIF demonstration: Adjust level labelsCorrection: To correct for multiple comparison testing and avoid Type I errors, different methods for correcting the p-value are available:
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bootstraps: When this option is selected, the bootstrapped marginal means are calculated. By default, the number of replications is set to 1000. This can be changed into the desired number. The simple main effects represent the effect of one independent variable for each level of the other independent variable, by conducting an ANOVA for each subset of the data as specified by the moderator variables.
GIF demonstration: Simple main effectsFactors: This box contains all the independent variables included in the analysis.
ANCOVA - dependent variable: - Cases: This column contains the independent variables, their interaction, the covariates, and the residual. - Sum of Squares: The summed squared group-mean differences. - df: Degrees of freedom of the model. - Mean Square: The estimate of population variance (the sum of squares divided by df's). - F: The value of the F-statistic. - p: The p-value. - VS-MPR: Vovk-Sellke Maximum p-ratio. - η2 : Estimated effect size eta-squared. - η2p : Estimated effect size partial eta-squared. - ω2 : Estimated effect size omega-squared.
Descriptives - dependent variable: - Independent variables: The levels of the independent variable(s) included in the analysis. If more than 1, the descriptives will be displayed for each combination of levels of the independent variables. - Mean: The mean per level or, if more than 1 independent variable, the mean per combination of levels. - SD: The standard deviation. - N: The sample size.
Test for Equality of Variances (Levene's): - F: F-statistic of Levene's test. - df1: Degrees of freedom calculated by k-1, where k represents the number of groups in the analysis. - df2: Degrees of freedom calculated by N-k-1, where N represents the total sample size, and k the number of groups in the analysis. - p: The p-value. If the p-value is significant, this means that the group variances of the dependent variable are not equal (i.e., the assumption of homogeneity is not met). - VS-MPR: Vovk-Sellke Maximum p-ratio.
Q-Q Plot: - With Q-Q plot the normality of the residuals can be inspected visually. The theoretical quantiles are displayed on the x-axis and standardized residuals on y-axis. The more dots are on the diagonal line, the more the data are normally distributed.
Deviation/Simple/Difference/Helmert/Repeated/Polynomial/Custom Contrast: - Comparison: The levels of the independent variable that are compared. For custom contrasts, the weight is displayed before each factor level name. - Estimate: The estimated mean difference between the compared levels. - SE: The standard error of the estimated mean. - df: The degrees of freedom of the model. - t: The value of the t-statistic. - p: The p-value. - % CI for Mean Difference: % confidence interval of the mean difference. This is 95% by default. - Lower: This is the lower bound of the confidence interval. - Upper: This is the upper bound of the confidence interval.
Post Hoc Comparisons (Standard)- independent variable: - The first two columns represent the levels/groups of the independent variable that are compared to each other. - Mean Difference: The mean difference between the levels. - % CI for Mean Difference: The confidence interval of the mean difference between the compared levels. By default this is set to 95%. - Lower: The lower bound of the confidence interval. - Upper: The upper bound of the confidence interval. - SE: The standard error of the mean difference. - t: The value of the t-statistic. - Cohen's d: The effect size Cohen's d. Cohen's d does not correct for multiple comparisons. - ptukey: Tukey's corrected p-value for multiple comparisons. - pscheffe: Scheffe's corrected p-value for multiple comparisons. - pbonf: Bonferroni's corrected p-value for multiple comparisons. - pholm: Holm's corrected p-value for multiple comparisons.
Games-Howell Post Hoc Comparisons - independent variable: - The first two columns represent the levels/groups of the independent variable that are compared to each other. - Mean Difference: The mean difference between the levels. - % CI for Mean Difference: The confidence interval of the mean difference between the compared levels. By default this is set to 95%. - Lower: The lower bound of the confidence interval. - Upper: The upper bound of the confidence interval. - SE: The standard error of the mean difference. - t: The value of the t-statistic. - ptukey: Tukey's corrected p-value for multiple comparisons.
Dunnett Post Hoc Comparisons: - The first two columns represent the levels/groups of the independent variable that are compared to each other. - Mean Difference: The mean difference between the levels. - % CI for Mean Difference: The confidence interval of the mean difference between the compared levels. By default this is set to 95%. - Lower: The lower bound of the confidence interval. - Upper: The upper bound of the confidence interval. - SE: The standard error of the mean difference. - t: The value of the t-statistic. - pdunnett: Dunnett's p-value.
Dunn's Post Hoc Comparisons: - The first two columns represent the levels/groups of the independent variable that are compared to each other. - z: The value for the z-statistic. - Wi: The mean ranking of the first level/group of the comparison. - Wj: The mean ranking of the second level/group of the comparison. - rrb: The value for the rank biserial correlation, based on pairwise Mann-Whitney U tests. - p: The p-value. - pbonf: Bonferroni's corrected p-value for multiple comparisons. - pholm: Holm's corrected p-value for multiple comparisons.
Bootstrapped Post Hoc Comparisons - Independent Variable: - The first two columns represent the levels/group of the independent variable that are compared to each other. - Mean Difference: The estimate of the mean difference between the levels. This estimate is based on the median of the bootstrap distribution. - Bias: The average difference between the bootstrapped mean differences and the estimated mean difference. - SE: The standard error of the bootstrapped mean differences. - 95% bca CI for Mean Difference: The bias corrected accelerated confidence interval of the mean difference between the compared levels. By default this is set to 95%. - Lower: The lower bound of the confidence interval. - Upper: The upper bound of the confidence interval.
Marginal Means - Independent Variable: - The first column contains the levels of the independent variable. - Marginal Mean: The marginal mean for each level of the independent variable. This mean is adjusted for all the other variables in the model. - SE: The standard error of the marginal mean. - Lower CI: The lower bound of the confidence interval. - Upper CI: The upper bound of the confidence interval. - t: The value for the t-statistic. - p: The p-value.
Bootstrapped Marginal Means - Independent Variable: - Independent variable: This column contains all the levels of the independent variables. - Marginal Mean: The estimate of the marginal mean for each level of the independent variable. This mean is adjusted for all the other variables in the model. The estimate is based on the median of the bootstrap distribution. - Bias: The average difference between the bootstrapped marginal mean and the estimated marginal mean. - SE: The standard error of the bootstrapped marginal means. - 95% bca CI for Mean Difference: The bias corrected accelerated confidence interval of the mean difference between the compared levels. By default this is set to 95%. - Lower: The lower bound of the confidence interval. - Upper: The upper bound of the confidence interval.
Simple Main Effects - Independent Variable: - The first column contains the levels of the other independent variable included in the analysis (if present). - Sum of Squares: The summed squared group-mean differences. - df: The degrees of freedom. - Mean Square: The estimate of the population variance (the sum of squares divided by the degrees of freedom) - F: The value of the F-statistic. - p: The p-value.
Kruskal-Wallis Test: - Factor: This column contains the independent variable included in the analysis. - Statistic: The value for the test statistic. - df: The degrees of freedom. - p: The p-value.
Independent variable on the x-axis and dependent variable on the y-axis. If other independent variables are included, either different lines representing different values of the other independent variable are displayed in the same plot, or different plots representing different values of the other independent variable are displayed.
Independent variable on the x-axis and dependent variable on the y-axis. If other independent variables are included, different plots representing different values of the other independent variable are displayed.
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