# Calypso Regression Details

## Contents

## Multiple linear regression

Multiple linear regression allows identification of complex associations between microbial community composition and multiple factors.

Example factors are BMI, age, blood sugar, ph, temperature or iron. In Calypso, regression models generally have the form:

dp = factor1 + factor2 + factor3 + ...

where dp is a dependent variable (such as taxa abundance or microbial diversity) and factor1 to factorN are multiple factors defined in the meta information file. In the regression analysis, the dependent variable is explained or "modeled" by the values of the included factors.

The following modes are available:

Type | Description |
---|---|

Taxa vs Factors | Identify associations between individual taxa and multiple factors. |

Time Series | Analysis of longitudinal data. |

Diversity vs Envp | Identify associations between community diversity and multiple factors. |

## Details

### Taxa vs Envp

Identify associations between individual taxa and all factors defined in the meta information file using multiple regression. For each taxon a regression model is fit of the form:

taxon = factor1 + factor2 + factor3 ...

All factors provided in the meta information file are included. The results of the analysis are shown in table format. For each taxon-factor pair model p-value (".p") and coefficient (".c") are shown, indicating if the factor-factor association is significant.

Use the drop-down menu to select a specific taxon. For each factor, two scatterplots are displayed. In the first plot, the values of the selected taxon are plotted versus the values of the factor. The Pearson correlation is given as R, the p-value indicates the significance of the correlation. In the second plot, the selected taxon is controlled by the remaining factors. Taxa are controlled by first fitting a multiple linear regression model including all remaining factors as explanatory variables. The residuals of this model are then plotted versus the selected taxon.

Multivariate paired data can be analyzed by mixed effect regression, which incorporates the paired variable (e.g. subject, animal or cage) as random effect and other factors (e.g. cases/controls or treatment) as fixed effect. These models can distinguish between group-specific effects (e.g. average in cases and controls) and subject or cage-specific effects. For paired data, select the "Paired" checkbox. A linear mixed effects model is fit of the form:

taxon = factor1 + factor2 + factor3 ... + pair1 + pair2 + ...,

where factors are included as fixed effects and pairs as random effect. The model is fit in R using the command: lmer(taxon ~ factor1 + factor2 + factor3 .. ... + (1|pair))

All factors defined in the meta information file are included. The pair information is taken from the third column of the meta information file.

Use the drop-down menu to select a specific taxon. For each factor f, two scatterplots are displayed. In the first plot, the values of the selected taxon are plotted versus factor f. In the second plot, the partial correlation is shown. The values of the selected taxon are controlled by all remaining factors (by fitting a regression model with the remaining factors and plotting the residuals versus f). The Pearson correlation is given as R, the p-value indicates the significance of the correlation.

For paired data select the "Paired" checkbox and press "Run Analysis". In the case of paired data, a linear mixed effects model is fit of the form:

taxon = fa1 + fa2 + fa3 ... + pair1 + pair2 + ...,

where the factors are included as fixed effects and the pairs as random effect.

### Time Series

Analysis of longitudinal data (time series). The secondary group of each sample is used as time point. For each taxon, a mixed effect regression model is fit of the form:

taxon = time point 1 + time point 2 + ...+ pair1 + pair2 + ...,

where time point is included as fixed effect and pair as random effect. This is calculated in R using the lmer() function via the formula:

taxon ~ as.factor(secondary group) + (1|pair).

Use the taxa drop-down menu to select a single taxon. A scatter plot is shown plotting the correlation between the selected taxon and each factor.

### Diversity vs Envp

Identify associations between community diversity and multiple factors. The diversity index can be set via the 'Index' drop down menu. Recommended indices are "Shannon" and "Richness". A regression model is fit of the form:

diversity = fa1 + fa2 + fa3 …

, where fa1, fa2, ... are factors.

The results of the regression are presented as table. For each factor the coefficient ("Estimate") and p-value are shown. P-values indicate significance of associations.

Additionally, two scatterplots are shown for each factor. The first plots community diversity vs the values of the factor. The p-value indicates the significance of the Pearson correlation between community diversity and the factor. In the second scatterplot, the diversity index is controlled for all remaining environmental variables.

For paired data, select the 'Paired' checkbox. The following regression model is fit for paired data:

difference in diversity (samples from same pair) = difference in fac1 + difference in fa2 + ...,

where fa1, fa2, ... are the environmental variables. The difference in diversity is computed for all combinations of samples from the same pair.

### Environmental variable

Two tables are shown. First, the correlation (Pearson correlation) between each taxa and the selected dependent variable. A p-value is given indicating the statistical significance of the correlation. The second table summaries the results of the fitted regression model. For each bacterial group, the coefficient as well as a p-value are given. The p-value indicates if the respective bacterial groups has a significant contribution to the fitted regression model.

Additionally a scatter plot is presented, which visualizes how well the fitted model explains the dependant variable. The fitted model can be used to predict the dependent variable given the explanatory variable. The scatterplot displays for each sample the observed/original value of dependence variable (as specified in the meta data file) versus the predicted value (the value of the dependent variable as predicted by the fit model). A p-value of the statistical significance of the fitted model is given.

Use the Taxa drop-down menu to select a specific taxon or OTU.Four scatterplots are shown:

The first (top left) visualized the dependency of the dependant variable from the abundance of the selected bacterial group. Depicted is the value of the dependent variable versus the abundance of the selected bacterial group.

The second plot (upper right) depicts the controlled dependent variable versus the abundance of the bacterial group. The dependent variable is controlled by first fitting a multiple linear regression model using all remaining bacterial groups as explanatory variables. Subsequently, the simple linear regression model is fit on the residuals of the multiple linear regression model.

The third plot (lower left) depicts the partial correlation between the dependent variable and the selected bacterial group. The controlled dependent variable is plotted versus the controlled abundance of the bacterial group.

The fourth figure (lower right) visualize how well the fit linear regression model explains the value of the dependant variable, or in other words how strongly the dependant variable depends on the abundance of the selected bacterial group. Depicted is the the predicted value of the dependant variable (predicted by the fit simple linear regression model) versus the observed/original value of the the dependant variable (as specified in the meta data file).