Question

phones	saturated	animal	deaths
124	33	8	81
49	31	6	55
181	38	8	80
4	17	2	24
22	20	4	78
152	39	6	52
75	30	7	88
54	29	7	45
43	35	6	50
41	31	5	69
17	23	4	66
22	21	3	45
16	8	3	24
10	23	3	43
63	37	6	38
170	40	8	72
125	38	6	41
12	25	4	38
221	39	7	52
171	33	7	52
97	38	6	66
254	39	8	89

a) First study the simple linear regression model for the heart attack death rates, on the only basis of the number of phones. Determine whether the number of phone is associated significantly with the heart attack death rate. b) Write the multiple linear regression model for the heart attack death rates on the basis of the number of phones and the proportion of saturated fat. Compute the associated least squares coefficient estimates. c) Test whether at least one of the predictors number of phones, or proportion of saturated fat, is useful in predicting the heart attack death rate. d) Compute the R2 statistic, and the residual standard error for the models in questions (b) and (c). Would you say that adding the proportion of saturated fat to the model significantly improves the accuracy? e) Write the multiple linear regression model for the heart attack rates on the basis of the number of phones, the proportion of saturated fat, and the proportion of animal fat. Compute the associated least squares coefficient. f) A country has the following features: 108 phones per 1000 inhabitants; 33% of saturated fat for men between the ages of 55 and 59; 7% of animal fat for men between the ages of 55 and 59. Predict the heart attack death rate for men between the ages of 55 and 59 in that country. g) Which coefficient estimates are significantly non-zero?

Answer 1

We may solve it via simple R-codes. To import the data, save the above table via copy-paste in a test file. Then, below is the R-command for import.

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> library(readr)
> dat <- read_delim("dat.txt", " ", escape_double = FALSE, trim_ws = TRUE)

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The above table is now a data-frame named dat in R.

(a) The regression equation to estimate would be $\dpi{80} \widehat{deaths} = \beta_0 &plus; \beta_1 * phones$ . The R-command would be as below.

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> summary(lm(dat$deaths ~ dat$phones))

Call:
lm(formula = dat$deaths ~ dat$phones)

Residuals:
    Min      1Q Median      3Q     Max
-24.157 -14.190 -2.637 12.219 32.762

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept) 46.23633    5.77111   8.012 1.14e-07 ***
dat$phones   0.12002    0.05044   2.379   0.0274 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 17.47 on 20 degrees of freedom
Multiple R-squared: 0.2206,   Adjusted R-squared: 0.1817
F-statistic: 5.662 on 1 and 20 DF, p-value: 0.02741

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As can be seen, the regression equation is now $\dpi{80} \widehat{deaths} = 46.23633 &plus; 0.12002 * phones$ .

Also, the phones variable is significantly associated to the deaths variable, as the null hypothesis $\dpi{80} H_0 : \beta_1 = 0$ would be rejected under usual 5% significance level (the p-value is 0.0274, which would be significant under 5% alpha level, but not under 1% alpha level).

(b) The regression equation to estimate would be $\dpi{80} \widehat{deaths} = \beta_0 &plus; \beta_1 * phones &plus; \beta_2 * saturated$ . The R-command would be as below.

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> summary(lm(dat$deaths ~ dat$phones + dat$saturated))

Call:
lm(formula = dat$deaths ~ dat$phones + dat$saturated)

Residuals:
    Min      1Q Median      3Q     Max
-22.311 -13.582 -3.174 14.342 32.402

Coefficients:
              Estimate Std. Error t value Pr(>|t|)
(Intercept)   35.55464   16.56005   2.147   0.0449 *
dat$phones     0.07895    0.07849   1.006   0.3271
dat$saturated 0.47073    0.68276   0.689   0.4989
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 17.7 on 19 degrees of freedom
Multiple R-squared: 0.2397,   Adjusted R-squared: 0.1596
F-statistic: 2.994 on 2 and 19 DF, p-value: 0.07407

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The regression equation would be hence $\dpi{80} \widehat{deaths} = 35.55464 &plus; 0.07895 * phones &plus; 0.47073 * saturated$ .

(c) The test of overall significance that $\dpi{80} H_0 : \beta_1 = \beta_2 = 0$ would have the F-statistic as $\dpi{80} F = \frac{R^2 / k}{(1 - R^2)/(n-k)}$ following the F distribution with k,n-k-1 degree of freedom. Here, n=22 and k=3 (number of parameters to be estimated). That is computed in the regression result in the last line as 2.994, having the p-value 0.07407. The test of overall significance would fail to reject the null hypothesis that $\dpi{80} H_0 : \beta_1 = \beta_2 = 0$ under usual 5% significance level. Hence, we may conclude that, with 95% confidence level, at least one of the explanatory variable - number of phones or proportion of saturated fat, is not useful in predicting the death rate.

(d) The R-square is given to be 0.2397 or 23.97%. The residual standard error would be $\dpi{80} RSE = \sqrt{\frac{RSS}{n-k}}$ , where RSS is the residual sum of square. It is given in the output as 17.7, but we may calculate it as by the R-commands below.

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> sqrt(sum(summary(lm(dat$deaths ~ dat$phones + dat$saturated))$resid^2)/19)
[1] 17.69967

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This would be approxed as 17.7, as given. The $resid beside the summary function would give the residuals, and those are squared, and then divided by degree of freedom n-k=22-3=19. The sqrt function is to calculate the square root.

(e) The regression equation to estimate would be $\dpi{80} \widehat{deaths} = \beta_0 &plus; \beta_1 * phones &plus; \beta_2 * saturated &plus; \beta_3*animal$ . The R-command would be as below.

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> summary(lm(dat$deaths ~ dat$phones + dat$saturated + dat$animal))

Call:
lm(formula = dat$deaths ~ dat$phones + dat$saturated + dat$animal)

Residuals:
    Min      1Q Median      3Q     Max
-24.134 -10.675 -1.435   9.321 29.800

Coefficients:
               Estimate Std. Error t value Pr(>|t|)
(Intercept)   23.999566 15.978866   1.502   0.1504
dat$phones    -0.006173   0.081298 -0.076   0.9403
dat$saturated -0.479869   0.757034 -0.634   0.5341
dat$animal     8.483500   3.846205   2.206   0.0406 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 16.13 on 18 degrees of freedom
Multiple R-squared: 0.4014,   Adjusted R-squared: 0.3017
F-statistic: 4.024 on 3 and 18 DF, p-value: 0.02357

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The regression would hence be as $\dpi{80} \widehat{deaths} = 23.999566 - 0.006173 * phones$ $\dpi{80} - 0.479869 * saturated &plus; 8.4835*animal$.

(f) The prediction for the given values would be $\dpi{80} \widehat{deaths} = 23.999566 - 0.006173 * 108 - 0.479869 *33 &plus; 8.4835*7$ or $\dpi{80} \widehat{deaths} = 66.8817$ , which is the heart attack death rate prediction for that age.

(g) Yet the test for overall significance is positive, meaning that the null that all the slope coefficients are equal to zero, can be rejected under 5% significance, not all the coefficients are significant individually.

As can be seen in the regression output, only animal's coefficient is significantly different from zero. Rest have p-values greater than 5% or 0.05, which is why we may say that their individual nulls $\dpi{80} H_0 : \beta_1 = 0$ and $\dpi{80} H_0 : \beta_2 = 0$ would be fail to be rejected.