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--- r:multiple_regression [2020/12/01 14:30] – [Multiple Regression] hkimscil
+++ r:multiple_regression [2023/10/19 08:23] (current) – hkimscil
@@ Line 19: / Line 19: @@
 </code>
 <code>
-three.predictor.model <- lm(ROLL ~ UNEM + HGRAD + INC, datavar)
-summary(two.predictor.model)
-two.predictor.model
 two.predictor.model <- lm(ROLL ~ UNEM + HGRAD, datavar)
 summary(two.predictor.model)
 two.predictor.model
+</code>
+<code>
+three.predictor.model <- lm(ROLL ~ UNEM + HGRAD + INC, datavar)
+summary(three.predictor.model)
+three.predictor.model
 </code>
-[[:sequential_regression#eg_3_college_enrollment_in_new_mexico_university|Sequential method]]
+<code>
+> two.predictor.model <- lm(ROLL ~ UNEM + HGRAD, datavar)
+> summary(two.predictor.model)
+Call:
+lm(formula = ROLL ~ UNEM + HGRAD, data = datavar)
+Residuals:
+    Min      1Q  Median      3Q     Max
+-2102.2  -861.6  -349.4   374.5  3603.5
+Coefficients:
+              Estimate Std. Error t value Pr(>|t|)
+(Intercept) -8.256e+03  2.052e+03  -4.023  0.00044 ***
+UNEM         6.983e+02  2.244e+02   3.111  0.00449 **
+HGRAD        9.423e-01  8.613e-02  10.941 3.16e-11 ***
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+Residual standard error: 1313 on 26 degrees of freedom
+Multiple R-squared:  0.8489,	Adjusted R-squared:  0.8373
+F-statistic: 73.03 on 2 and 26 DF,  p-value: 2.144e-11
+> two.predictor.model
+Call:
+lm(formula = ROLL ~ UNEM + HGRAD, data = datavar)
+Coefficients:
+(Intercept)         UNEM        HGRAD
+ -8255.7511     698.2681       0.9423
+>
+</code>
+<code>
+> three.predictor.model <- lm(ROLL ~ UNEM + HGRAD + INC, datavar)
+> summary(three.predictor.model)
+Call:
+lm(formula = ROLL ~ UNEM + HGRAD + INC, data = datavar)
+Residuals:
+     Min       1Q   Median       3Q      Max
+-1148.84  -489.71    -1.88   387.40  1425.75
+Coefficients:
+              Estimate Std. Error t value Pr(>|t|)
+(Intercept) -9.153e+03  1.053e+03  -8.691 5.02e-09 ***
+UNEM         4.501e+02  1.182e+02   3.809 0.000807 ***
+HGRAD        4.065e-01  7.602e-02   5.347 1.52e-05 ***
+INC          4.275e+00  4.947e-01   8.642 5.59e-09 ***
+---
+Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
+Residual standard error: 670.4 on 25 degrees of freedom
+Multiple R-squared:  0.9621,	Adjusted R-squared:  0.9576
+F-statistic: 211.5 on 3 and 25 DF,  p-value: < 2.2e-16
+> three.predictor.model
+Call:
+lm(formula = ROLL ~ UNEM + HGRAD + INC, data = datavar)
+Coefficients:
+(Intercept)         UNEM        HGRAD          INC
+ -9153.2545     450.1245       0.4065       4.2749
+>
+</code>
+만약에
+  * unemployment rate (UNEM) = 9%, 12%, 3%
+  * spring high school graduating class (HGRAD) = 100000, 98000, 78000
+  * a per capita income (INC) of \$30000, \$28000, \$36000
+  * 일 때, enrollment는 어떻게 predict할 수 있을까?
+위에서 얻은 prediction model은 아래와 같다.
+$$ \hat{Y} = -9153.2545 + 450.1245 \cdot UNEM + 0.4065 \cdot HGRAD + 4.2749 \cdot INC $$
+여기에 위의 정보를 대입해 보면 된다.
+<code>
+new.data <- data.frame(UNEM=c(9, 12, 3), HGRAD=c(100000, 98000, 78000), INC=c(30000, 28000, 36000))
+predict(three.predictor.model, newdata=new.data)
+</code>
+<code>
+> new.data <- data.frame(UNEM=c(9, 10, 15), HGRAD=c(100000, 98000, 78000), INC=c(30000, 28000, 19000))
+> predict(three.predictor.model, newdata=new.data)
+        2        3
+.0 154879.4 110526.6
+>
+</code>
+\begin{align*}
+\hat{Y} & = -9153.2545 + 450.1245 \cdot \text{UNEM} + 0.4065 \cdot \text{HGRAD} + 4.2749 \cdot \text{INC}  \\
+.0 & = -9153.2545 + 450.1245 \cdot (9) + 0.4065 \cdot (100000) + 4.2749 \cdot (30000) \\
+.4 & = -9153.2545 + 450.1245 \cdot (10) + 0.4065 \cdot (98000) + 4.2749 \cdot (28000) \\
+.6 & = -9153.2545 + 450.1245 \cdot (15) + 0.4065 \cdot (78000) + 4.2749 \cdot (19000) \\
+\end{align*}
+beta coefficient 살펴보기
+see [[:beta coefficients]]
+<code>
+# install.packages('lm.beta')
+# library(lm.beta)
+lm.beta(three.predictor.model)
+</code>
+<code>
+> # install.packages('lm.beta')
+> # library(lm.beta)
+> lm.beta(three.predictor.model)
+Call:
+lm(formula = ROLL ~ UNEM + HGRAD + INC, data = datavar)
+Standardized Coefficients::
+(Intercept)        UNEM       HGRAD         INC
+.0000000   0.1553619   0.3656177   0.6061762
+>
+</code>
+by hand
+<code>
+# coefficient * (sd(x)/sd(y)) 이므로
+#
+attach(datavar)
+sd.roll <- sd(ROLL)
+sd.unem <- sd(UNEM)
+sd.hgrad <- sd(HGRAD)
+sd.inc <- sd(INC)
+b.unem <- three.predictor.model$coefficients[2]
+b.hgrad <- three.predictor.model$coefficients[3]
+b.inc <- three.predictor.model$coefficients[4]
+## or
+b.unem <- 4.501e+02
+b.hgrad <- 4.065e-01
+b.inc <- 4.275e+00
+b.unem * (sd.unem / sd.roll)
+b.hgrad * (sd.hgrad / sd.roll)
+b.inc * (sd.inc / sd.roll)
+lm.beta(three.predictor.model)
+</code>
+output of the above
+<code>
+> sd.roll <- sd(ROLL)
+> sd.unem <- sd(UNEM)
+> sd.hgrad <- sd(HGRAD)
+> sd.inc <- sd(INC)
+>
+> b.unem <- three.predictor.model$coefficients[2]
+> b.hgrad <- three.predictor.model$coefficients[3]
+> b.inc <- three.predictor.model$coefficients[4]
+>
+> ## or
+> b.unem <- 4.501e+02
+> b.hgrad <- 4.065e-01
+> b.inc <- 4.275e+00
+>
+>
+> b.unem * (sd.unem / sd.roll)
+[1] 0.1554
+> b.hgrad * (sd.hgrad / sd.roll)
+[1] 0.3656
+> b.inc * (sd.inc / sd.roll)
+[1] 0.6062
+>
+> lm.beta(three.predictor.model)
+Call:
+lm(formula = ROLL ~ UNEM + HGRAD + INC, data = datavar)
+Standardized Coefficients::
+(Intercept)        UNEM       HGRAD         INC
+.0000      0.1554      0.3656      0.6062
+>
+</code>
+see also [[:sequential_regression#eg_3_college_enrollment_in_new_mexico_university|Sequential method]] regression modeling by hand
+see also [[:statistical regression methods]] regression modeling by computing
+<code>
+> fit <- three.predictor.model
+> step <- stepAIC(fit, direction="both")
+Start:  AIC=381.2
+ROLL ~ UNEM + HGRAD + INC
+        Df Sum of Sq      RSS AIC
+<none>               11237313 381
+- UNEM   1   6522098 17759411 392
+- HGRAD  1  12852039 24089352 401
+- INC    1  33568255 44805568 419
+>
+</code>
 ====== Housing ======
 {{housing.txt}}
@@ Line 37: / Line 239: @@
 ====== etc ======
+{{:marketing_from_datarium.csv}}
 <code>
+marketing <- read.csv("http://commres.net/wiki/_media/marketing_from_datarium.csv")
+</code>
+<code>
+# install.packages("tidyverse", dep=TRUE)
 library(tidyverse)
 data("marketing", package = "datarium")
@@ Line 45: / Line 253: @@
   * Note that to list all the independent (explanatory) variables, you could use ''lm (sales ~ ., data="marketing")''.
   * You could also use ''-'' sign to subtract ivs. ''lm(sales ~ . - newspapers, data = "marketing")''
 <code>
@@ Line 271: / Line 480: @@
 | interest  | 1 (a)  | 894463 (1)  | 894463  | 179.6471179  |
 | unemp  | 1 (b)  | 22394 (2) | 22394  | 4.497690299  |
-| res  | 21 (c)  | 104559 (3) | 4979  |   |
+| res  | 21 %%(%%c%%)%%  | 104559 (3) | 4979  |   |
 | total  | 23  | 1021416 (4)  |   |   |
-| interst \\ + enemp  |   | 916857 (5)  |   |   |
+| interest \\ + enemp  |   | 916857 (5)  |   |   |
 (4) = (1) + (2) + (3)