Question

Construct a regression model for predicting total charges from length of stay for DRG 105.

a. State the null and alternative hypotheses and alpha level.

b. Prepare a scatter diagram with the regression line for the two variables.

c. What are the r and r2? What is the importance of the r and r2 results?

d. What is the regression equation?

e. What are your conclusions?

     Gender     Age     LOS     Charges    Payor

1    Female     47        20 $91,683    Medicaid

2    Female     75        43       $93,708    Medicare

3    Female 84          7      $21,446     Medicare

4    Female     50         13 $37,797 Medicare

5    Male 77         14      $54,364    Medicare

6    Male    57           4     $17,626    Medicare

7    Male        73           4     $12,832      Medicare

8 Female     56 1     $36,153      Medicaid

9    Male 69           1 $14,907      Medicaid

10    Female   81    23   $104,148      Medicare

11    Male       21            5 $21,423    Medicaid

12    Female   37            5      $24,971     Medicaid

13    Female   69            4 $17,022    Medicare

14 Female    89           17      $50,652 Medicare

15   Male        28 35      $186,496    Medicaid

16    Male       47             6      $24,441      Medicaid

17   Male        87            11     $35,349    Medicare

18   Female     85             5 $22,155      Medicare

19    Male      56               5     $24,455      Managed Care

20    Male    45             11      $36,401 Medicaid

21    Male      82    6    $25,783      Medicare

22    Female 65             10     $37,055       Managed Care

23    Male     67             4      $19,236       Medicare

24    Male       59           23       $60,132       Other

25    Female    67            7       $35,777       Medicare

26    Male        53 4       $19,972       Managed Care

27    Male    71           7 $25,409      Medicare

28    Female   79            6       $281,140     Medicare

29    Male 63    1        $41,283     Medicaid

30    Male       53           19        $71,439     Medicaid

31    Female    75            9        $33,735 Medicare

32    Female    68            9        $37,830     Gov Mngd Care

33    Male        37 4          $22,311     Medicaid

Total N 33        33          33            33              33

Answer 1

I used R software to solve this question.

R codes and output:

> d=read.table('data.csv',header=T,sep=',')

> head(d)

LOS Charges

1 20 $91,683

2 43 $93,708

3   7 $21,446

4 13 $37,797

5 14 $54,364

6   4 $17,626

> attach(d)

The following objects are masked from d (pos = 3):

    Charges, LOS

> Charge=as.numeric(Charges)

> fit=lm(Charge~LOS)

> summary(fit)

Call:

lm(formula = Charge ~ LOS)

Residuals:

    Min      1Q Median      3Q     Max

-20.850 -5.925   1.690   7.152 13.614

Coefficients:

            Estimate Std. Error t value Pr(>|t|)   

(Intercept) 13.0014     2.3517   5.529 4.72e-06 ***

LOS           0.3847     0.1675   2.297   0.0285 *

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 9.082 on 31 degrees of freedom

Multiple R-squared: 0.1454, Adjusted R-squared: 0.1179

F-statistic: 5.275 on 1 and 31 DF, p-value: 0.02855

> plot(LOS, Charge)

> abline(fit)

> cor(Charge,LOS)

[1] 0.3813442

Que.a

Hypothesis:

The fitted regression model is not statistically significant.

The fitted regression model is statistically significant.

Que.b

Scatter plot:

Que.c

Correlation coefficient = r = 0.3813442

Coefficient of determination = r2 = 0.1454

Correlation coefficient (r) gives the strength of linear relationship between two variables whereas Coefficient of determination tells information about how much variation in one variable is explained by the other variable.

Que.d

Regression equation:

Total charges = 13.0014 + 0.3847 LOS

Que.e

Since p-value for t test ( for testing significance of slope ) is 0.0285, which is less than 0.05, hence we reject null hypothesis and conclude that fitted regression model is statistically significant.