Question

What are the various components of the output and how is the output of value to understanding data in regression analysis?

Answer 1

Solution:

The ability to determine model fit is a tricky process. The metrics used to determine model fit can have different values based on the type of data. Hence, we need to be extremely careful while interpreting regression analysis. Following are some metrics you can use to evaluate your regression model:

    R Square (Coefficient of Determination) - As explained above, this metric explains the percentage of variance explained by covariates in the model. It ranges between 0 and 1. Usually, higher values are desirable but it rests on the data quality and domain. For example, if the data is noisy, you'd be happy to accept a model at low R² values. But it's a good practice to consider adjusted R² than R² to determine model fit.
    Adjusted R²- The problem with R² is that it keeps on increasing as you increase the number of variables, regardless of the fact that the new variable is actually adding new information to the model. To overcome that, we use adjusted R² which doesn't increase (stays same or decrease) unless the newly added variable is truly useful.
    F Statistics - It evaluates the overall significance of the model. It is the ratio of explained variance by the model by unexplained variance. It compares the full model with an intercept only (no predictors) model. Its value can range between zero and any arbitrary large number. Naturally, higher the F statistics, better the model.
    RMSE / MSE / MAE - Error metric is the crucial evaluation number we must check. Since all these are errors, lower the number, better the model. Let's look at them one by one:
        MSE - This is mean squared error. It tends to amplify the impact of outliers on the model's accuracy. For example, suppose the actual y is 10 and predictive y is 30, the resultant MSE would be (30-10)² = 400.
        MAE - This is mean absolute error. It is robust against the effect of outliers. Using the previous example, the resultant MAE would be (30-10) = 20
        RMSE - This is root mean square error. It is interpreted as how far on an average, the residuals are from zero. It nullifies squared effect of MSE by square root and provides the result in original units as data.

Let's understand the regression output in detail:

    Intercept - This is the βo value. It's the prediction made by model when all the independent variables are set to zero.
    Estimate - This represents regression coefficients for respective variables. It's the value of slope. Let's interpret it for Chord_Length. We can say, when Chord_Length is increased by 1 unit, holding other variables constant, Sound_pressure_level decreases by a value of -35.69.
    Std. Error - This determines the level of variability associated with the estimates. Smaller the standard error of an estimate is, more accurate will be the predictions.
    t value - t statistic is generally used to determine variable significance, i.e. if a variable is significantly adding information to the model. t value > 2 suggests the variable is significant. I used it as an optional value as the same information can be extracted from the p value.
    p value - It's the probability value of respective variables determining their significance in the model. p value < 0.05 is always desirable.