# Regression - Linear Regression

The Linear Regression models the linear relationship between a dependent variable and one or more independent variables. The linear regression option is most commonly used when the dependent variable is continuous.

## Interpretation

Variable statistics measure the impact and significance of individual variables within a model, while overall statistics apply to the model as a whole. Both are shown in the regression output.

### Variable statistics

Estimate the magnitude of the coefficient indicates the size of the change in the independent variable as the value of the dependent variable changes. A positive number indicates a direct relationship (y increases as x increases), and a negative number indicates an inverse relationship (y decreases as x increases.

The coefficient is colored if the variable is statistically significant at the 5% level.

Standard Error measures the accuracy of an estimate. The smaller the standard error, the more accurate the predictions.

t-statistic the estimate divided by the standard error. The magnitude (either positive or negative) indicates the significance of the variable. The values are highlighted based on their magnitude.

p-value expresses the t-statistic as a probability. A p-value under 0.05 means that the variable is statistically significant at the 5% level; a p-value under 0.01 means that the variable is statistically significant at the 1% level. P-values under 0.05 are shown in bold.

### Overall statistics

n the sample size of the model

R-squared assess the goodness of fit of the model. A larger number indicates that the model captures more of the variation in the dependent variable.

AIC Akaike Information Criterion is a measure of the quality of the model. When comparing similar models, the AIC can be used to identify the superior model.

## Example

The example below is a model that predicts a survey respondent’s average monthly spending on fast-food products based on characteristics like age, gender, and work status.

### Create a Linear Regression Model in Displayr

1. Go to Insert > Regression > Linear Regression
2. Under Inputs > Outcome, select your dependent variable
3. Under Inputs > Predictor(s), select your independent variables

## Object Inspector Options

Outcome The variable to be predicted by the predictor variables.

Predictors The variable(s) to predict the outcome.

Algorithm The fitting algorithm. Defaults to Regression but may be changed to other machine learning methods.

Type: You can use this option to toggle between different types of regression models, but note that certain types are not appropriate for certain types of outcome variable. The other types are not appropriate for a continuous outcome variable.

Linear.
Binary Logit See Regression - Binary Logit.
Ordered Logit See Regression - Ordered Logit.
Multinomial Logit See Regression - Multinomial Logit.
Poisson See Regression - Poisson Regression.
Quasi-Poisson See Regression - Quasi-Poisson Regression.
NBD See Regression - NBD Regression.

Robust standard errors Computes standard errors that are robust to violations of the assumption of constant variance (i.e., heteroscedasticity). See Robust Standard Errors. This is only available when Type is Linear.

Missing data See Missing Data Options.

Output

Summary The default; as shown in the example above.
Detail Typical R output, some additional information compared to Summary, but without the pretty formatting.
ANOVA Analysis of variance table containing the results of Chi-squared likelihood ratio tests for each predictor.
Relative Importance Analysis The results of a relative importance analysis. See here and the references for more information. This option is not available for Multinomial Logit. Note that categorical predictors are not converted to be numeric, unlike in Driver (Importance) Analysis - Relative Importance Analysis.
Shapley regression See here and the references for more information. This option is only available for Linear Regression. Note that categorical predictors are not converted to be numeric, unlike in Driver (Importance) Analysis - Shapley.
Effects Plot Plots the relationship between each of the Predictors and the Outcome. Not available for Multinomial Logit.

Correction The multiple comparisons correction applied when computing the p-values of the post-hoc comparisons.

Variable names Displays Variable Names in the output instead of labels.

Absolute importance scores Whether the absolute value of Relative Importance Analysis scores should be displayed.

Auxiliary variables Variables to be used when imputing missing values (in addition to all the other variables in the model).

Weight. Where a weight has been set for the R Output, it will automatically applied when the model is estimated. By default, the weight is assumed to be a sampling weight, and the standard errors are estimated using Taylor series linearization (by contrast, in the Legacy Regression, weight calibration is used). See Weights, Effective Sample Size and Design Effects.

Filter The data is automatically filtered using any filters prior to estimating the model.

Crosstab Interaction Optional variable to test for interaction with other variables in the model. The interaction variable is treated as a categorical variable. Coefficients in the table are computed by creating separate regressions for each level of the interaction variable. To evaluate whether a coefficient is significantly higher (blue) or lower (red), we perform a t-test of the coefficient compared to the coefficient using the remaining data as described in Driver Analysis. P-values are corrected for multiple comparisons across the whole table (excluding the NET column). The P-value in the sub-title is calculated using a the likelihood ratio test between the pooled model with no interaction variable, and a model where all predictors interact with the interaction variable.

Random seed Seed used to initialize the (pseudo)random number generator for the model fitting algorithm. Different seeds may lead to slightly different answers, but should normally not make a large difference.

Additional options are available by editing the code.

When using this feature you can obtain additional information that is stored by the R code which produces the output.

1. To do so, select Create > R Output.
2. In the R CODE, paste: item = YourReferenceName
3. Replace YourReferenceName with the reference name of your item. Find this in the Report tree or by selecting the item and then going to Properties > General > Name from the object inspector on the right.
4. Below the first line of code, you can paste in snippets from below or type in str(item) to see a list of available information.

For a more in depth discussion on extracting information from objects in R, checkout our blog post here.

Properties which may be of interest are:

• Summary outputs from the regression model:
item\$summary\$coefficients # summary regression outputs