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If one is measuring safety climate (which has dimensions or contributing factors to the overall safety climate), what do the results of a regression tell you? I know it either states if they are related as in being statistically significant or not, but how?

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First of all we'll need to be sure that the model is well specified, assumptions are checked and not grossly violated, influential data point or other anomalies are addressed...etc. Then we would look at the regression output. If you are not sure about the above steps, get statistical help because the output of an atrocious model and that of a wonderful model can look very similar to untrained eyes.

Resources on how to appreciate regression output are plenty online. It'd also help if you know which software you'll be using because while they are all regressions, the outputs do look a bit different albeit containing mostly the same information.

Here are a few that may help you to get started: 1, 2, 3, 4.

In addition, consider reading up on some applied statistics or biostatistics text chapters. Usually in 1-2 chapters we can glean some ideas on how to appreciate the regression outputs.

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Please take my answer critically until someone more experienced than I answers. :)

Your multiple regression model will fit the line that produces the smallest squared error between your Xs and your y. That way, each fitted parameter in your model can be interpreted as the influence of its feature on the outcome.

For instance if your fitted model gives you: y = intercept + theta0 * x0 + theta1 * x1

Then you can interpret theta0 as "If we assume the relationship betwene x0 and y is linear, and holding x1 constant, then for every unit of x0, y would increase by theta0."

You can also include quadratic terms, such as x0 squared, to describe non-linear relationships.

The P-value of each parameter tells you to which degree you can be confident that it is significant: with a P-value of 0.05, you are 95% confident that the feature has a positive or negative relationship with y given the data included in the model. You can see this more concretely in the computed confidence intervals.

The R^2 tells you how much of the variance of y is explained by your features.

You can only interpret the parameters and P-values of your model if the underlying hypotheses of the regression are verified: linear relationship, low correlation between your features... Take a look here for an exhaustive list.

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