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How to do multiple linear regression analysis with JMP?

What is Multiple Linear Regression ?

Multiple linear regression is used to explore Multiple independent and dependent variables The relationship between, and used to simulate the relationship between the reflection variable and two or more dependent variables.

use JMP conduct Multiple Linear Regression

Take the data in Figure 1 as an example to explore possible influencing factors of physical fitness scores.

figure 1

Click the Analysis menu in JMP → " fit model "(Figure 2), import the physical health score into "Y" in the pop-up dialog box " , import the variables expected to be subjected to a multivariate analysis into the Construction Model Effects (Figure 3).

figure 2

image 3 " The second step of the operation diagram of Fitting Model

As we can see in Figure 3, there are two option boxes on the upper right side of the dialog box, namely "Trait" and "Key Points".

When the dependent variable is a continuous variable (At this time, the left side of the variable name is displayed as a blue triangle), the "Character" is automatically set to "Standard Least Squares", that is using linear regression .

" focus " The default is the effect leverage ratio, you can choose other, it does not affect the result , but the content displayed in the result is different, some show more, some show less, and you can adjust the content to be displayed in the result by yourself, so the option "Key" can be defaulted.

02 Interpretation of input results

The default output of linear regression includes the results of several different parts, and only the important parts are introduced below.

Part 1 Leverage Chart

The first part is Leverage chart, also known as partial regression residual leverage chart, or adding variable chart, which reflects the relationship between an independent variable and a dependent variable after deducting the influence of other factors.

In this case, the multivariate analysis included 6 independent variables, so the results presented a plot of the deviation of the 6 independent variables from the outcome after adjusting for the other 5 independent variables. Due to limited space, this paper only presents the leverage plots for the first 3 independent variables (Figure 4).

Figure 4 Leverage chart results (partial display)

Leverage plots are not the same for continuous and categorical variables.

Continuous variables are easy to understand, such as the lever plot of age, which is negatively correlated with physical fitness scores after adjusting for other factors;
Results for categorical variables can be obtained by Least Squares Mean ” to observe the difference between two or more classes. As shown in the table for drinking, after adjusting for other factors, the least squares mean of physical health scores for drinking and non-drinking were 48.1 and 45.4, respectively, which is not equivalent to the conventional mean.

Part 2 " Predictive value - actual value " picture

The second part is the "predicted value - actual value" graph (Figure 5), The figure reflects the degree of agreement between the predicted value of the model and the actual observed value. The abscissa is the predicted value, and the ordinate is the actual value. The closer the two are. 45 The degree line, the better the model fitting effect.

In this example R² =0.42, which can be considered as a good regression model in the medical field.

Figure 5 " Predicted Value - Actual Value" graph

Part 3 Effect summary results

The third part is the effect summary results (Figure 6), This part reflects the contribution value of the respective variables, mainly through LogWorth value to reflect, LogWorth value equal to -log₁₀ P .

As can be seen from Figure 6, the largest contribution of the 6 independent variables is cardiac function class and dyspnea, while age, BMI, alcohol consumption and smoking have less influence. The results can intuitively reflect the contribution of independent variables, and can also be used as a reference for further variable screening.

Figure 6 Summary of Effects

Part 4 " Predictive value - residual " picture

The fourth part is the "predicted value-residuals" graph (Fig. 7), which is a scatter plot with the predicted value of the variable as the abscissa and the residual as the ordinate.

The residual plot can indicate whether the model fit is reasonable, if the residuals are randomly distributed in the y=0 If it is a non-random distribution (such as showing a specific change trend), it means that there may be a problem with the model, and it is necessary to reconsider whether the preconditions for model construction are satisfied.

The results of this example show that the distribution of residuals is relatively uniform, and no obvious change trend is found. It can be considered that the model fits the data well.

Figure 7 " Predicted Values - Residuals" Plot

Part 5 parameter estimates

The fifth part is the parameter estimates (Figure 8), which The part is often the result that we are more concerned about, and gives the parameter estimate, standard error, t value and P value (i.e. in the result " probability >|t|” ) .

For continuous variables, the parameter estimates are only 1 row of results; for categorical variables, the number of rows of results presented is the number of categories-1, showing results compared to the reference category, such as "heart function class" It is a 4-category, so 4-1=3 rows of results are displayed, which are the results of comparing cardiac function classes 1, 2, 3 and 4 (categories that do not appear in the results).

Figure 8 Parameter estimates

Note, however, that JMP The default parameter estimates in categorical variables in are "effect codes", not dummy variable codes. In general, we are looking at categorical variables and expect to see dummy variables encoded, in this case the opposite.

The difference between effect coding and dummy variable coding is that dummy variables reflect Differences between other classes and the reference class ; the effect code reflects the The difference between each class and the mean . Therefore, when there are categorical variables in the independent variables, do not use the results of the parameter estimates, but choose the results of the parameterized estimates of the indicators.

In order to output our more accustomed dummy variable code, we need to click on the red down arrow to the left of the response "physical fitness score" → "estimated value" → " Parameterized estimates of indicators” (Figure 9) .

Figure 9 " Parameterized Estimated Values of Indicators "Operation Steps"

At this time, the result will give the result of the parameterization of the indicator function, and the parameter estimation result corresponding to the categorical variable is the result we really hope to get (Figure 10).

Figure 10 Parameterized estimates of indicators

Figure 10 shows the parameterization results of the index function (taking cardiac function class as an example), compared with cardiac function class 4 (the category not shown in the results is the reference class), the physical health score of cardiac function class 1 is better than that of cardiac function class 4 The average height was 36.29, and the heart function class 2 was 22.22 higher than the heart function class 4 on average, and the heart function class 3 was 11.54 higher than the heart function class 4 on average.

Comparing the results in Figure 10 and Figure 8, it can be seen that the continuous variables of the two results are the same, but the categorical variables are different. Therefore, once again, If the independent variables have categorical variables, the indicator function must be selected to parameterize the results . The above is how to use JMP shared today A multivariate analysis of linear regression was performed.