Linear regression Model
Simple linear regression model
In univariate regression model, you can use scatter plot to visualize model. For example, you can make simple linear regression model with data radial included in package moonBook. The radial data contains demographic data and laboratory data of 115 patients performing IVUS(intravascular ultrasound) examination of a radial artery after tansradial coronary angiography. The NTAV(normalized total atheroma volume measured by intravascular ultrasound(IVUS) in cubic mm) is a quantitative measurement of atherosclerosis. Suppose you want to predict the amount of atherosclerosis(NTAV) from age.
Loading required package: moonBook
Call:
lm(formula = NTAV ~ age, data = radial)
Residuals:
Min 1Q Median 3Q Max
-45.231 -14.626 -4.803 9.685 100.961
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 44.3398 14.6251 3.032 0.00302 **
age 0.3848 0.2271 1.694 0.09302 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 23.94 on 113 degrees of freedom
Multiple R-squared: 0.02477, Adjusted R-squared: 0.01614
F-statistic: 2.87 on 1 and 113 DF, p-value: 0.09302You can get the regression equation from summary of regression model:
y=0.38*x+44.34You can visualize this model easily with ggplot2 package.
You can make interactive plot easily with ggPredict() function included in ggiraphExtra package.
With this plot, you can identify the points and see the regression equation with your mouse.Multiple regression model without interaction
You can make a regression model with two predictor variables. Now you can use age and sex as predictor variables.
Call:
lm(formula = NTAV ~ age + sex, data = radial)
Residuals:
Min 1Q Median 3Q Max
-46.025 -12.687 -1.699 5.784 89.419
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 17.8697 14.3846 1.242 0.21673
age 0.6379 0.2134 2.989 0.00344 **
sexM 20.5476 4.1943 4.899 3.27e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 21.82 on 112 degrees of freedom
Multiple R-squared: 0.1969, Adjusted R-squared: 0.1825
F-statistic: 13.73 on 2 and 112 DF, p-value: 4.659e-06From the result of regression analysis, you can get regression regression equations of female and male patients :
For female patient, y=0.64*x+17.87
For male patient, y=0.64*x+38.42You can visualize this model with ggplot2 package.
equation1=function(x){coef(fit1)[2]*x+coef(fit1)[1]}
equation2=function(x){coef(fit1)[2]*x+coef(fit1)[1]+coef(fit1)[3]}
ggplot(radial,aes(y=NTAV,x=age,color=sex))+geom_point()+
stat_function(fun=equation1,geom="line",color=scales::hue_pal()(2)[1])+
stat_function(fun=equation2,geom="line",color=scales::hue_pal()(2)[2])
You can make interactive plot easily with ggPredict() function included in ggiraphExtra package.
Multiple regression model with interaction
You can make a regession model with two predictor variables with interaction. Now you can use age and DM(diabetes mellitus) and interaction between age and DM as predcitor variables.
Call:
lm(formula = NTAV ~ age * DM, data = radial)
Residuals:
Min 1Q Median 3Q Max
-44.094 -15.115 -4.093 9.102 102.024
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 49.6463 16.8660 2.944 0.00395 **
age 0.2925 0.2648 1.105 0.27174
DM -20.8618 34.8936 -0.598 0.55115
age:DM 0.3453 0.5353 0.645 0.52026
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 24.1 on 111 degrees of freedom
Multiple R-squared: 0.0292, Adjusted R-squared: 0.002966
F-statistic: 1.113 on 3 and 111 DF, p-value: 0.347The regression equation in this model are as follows: For patients without DM(DM=0), the intercept is 49.65 and the slope is 0.29. For patients with DM(DM=1), the intercept is 49.65-20.86 and the slope is 0.29+0.35.
For patients without DM(DM=0), y=0.29*x+49.65
For patients without DM(DM=1), y=0.64*x+28.78You can visualize this model with ggplot2.
You can make interactive plot easily with ggPredict() function included in ggiraphExtra package.
Multiple regression model with two continuous predictor variables with or without interaction
You can make a regession model with two continuous predictor variables. Now you can use age and weight(body weight in kilogram) as predcitor variables.
Call:
lm(formula = NTAV ~ age * weight, data = radial)
Residuals:
Min 1Q Median 3Q Max
-48.482 -13.815 -2.079 6.886 93.187
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 37.60533 111.46748 0.337 0.736
age -0.32698 1.69737 -0.193 0.848
weight -0.10416 1.74620 -0.060 0.953
age:weight 0.01468 0.02687 0.546 0.586
Residual standard error: 22.91 on 111 degrees of freedom
Multiple R-squared: 0.1222, Adjusted R-squared: 0.09851
F-statistic: 5.152 on 3 and 111 DF, p-value: 0.00226From the analysis, you can get the regression equation for a patient with body weight 40kg, the intercept is 37.61+(-0.10416)*40 and the slope is -0.33+0.01468*40
For bodyweight 40kg, y=0.26*x+33.44
For bodyweight 50kg, y=0.41*x+32.4
For bodyweight 90kg, y=0.99*x+28.23To visualize this model, the simple ggplot command shows only one regression line.
You can easily show this model with ggPredict() function.
Multiple regression model with three predictor variables
You can make a regession model with three predictor variables. Now you can use age and weight(body weight in kilogram) and HBP(hypertension) as predcitor variables.
Call:
lm(formula = NTAV ~ age * weight * HBP, data = radial)
Residuals:
Min 1Q Median 3Q Max
-43.453 -14.125 -3.226 7.724 88.126
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 64.11678 155.82328 0.411 0.682
age -0.67650 2.47339 -0.274 0.785
weight -0.39685 2.37886 -0.167 0.868
HBP -101.94261 238.52253 -0.427 0.670
age:weight 0.01686 0.03804 0.443 0.658
age:HBP 1.27972 3.64467 0.351 0.726
weight:HBP 1.52494 3.75529 0.406 0.685
age:weight:HBP -0.01666 0.05777 -0.288 0.774
Residual standard error: 22.8 on 107 degrees of freedom
Multiple R-squared: 0.1626, Adjusted R-squared: 0.1078
F-statistic: 2.967 on 7 and 107 DF, p-value: 0.006982From the analysis result, you can get the regression equation for a patient without hypertension(HBP=0) and body weight 60kg: the intercept is 64.12+(-0.39685*60) and the slope is -0.67650+(0.01686*60). The equation for a patient with hypertension(HBP=1) and same body weight: the intercept is 64.12+(-0.39685*60-101.94) and the slope is -0.67650+(0.01686*60)+1.27972+(-001666*60).
To visualize this model, you can make a faceted plot with ggPredict() function. You can see the regression equation of each subset with hovering your mouse on the regression lines.