Homework 3 Answers

BSTA 512/612

Due: Friday February 14, 2025 at 11pm

Answers are not necessarily complete! This is just meant to serve as a check if you are stuck.

Questions

Question 1

This question and data are adapted from this textbook.

In an experiment designed to describe the dose–response curve for vitamin K, individual rats were depleted of their vitamin K reserves and then fed dried liver for 4 days at different dosage levels. The response of each rat was measured as the concentration of a clotting agent needed to clot a sample of its blood in 3 minutes. The results of the experiment on 12 rats are given in the following table; values are expressed in common logarithms for both dose and response.

clot = read_excel(here("data/CH05Q09.xls"))
clot %>% gt() %>%
  cols_label(RAT = md("**Rat**"),
             LOGCONC = md("**Log10 Concentration (Y)**"),
             LOGDOSE = md("**Log10 Dose (X)**"))

Rat	Log10 Concentration (Y)	Log10 Dose (X)
1	2.65	0.18
2	2.25	0.33
3	2.26	0.42
4	1.95	0.54
5	1.72	0.65
6	1.60	0.75
7	1.55	0.83
8	1.32	0.92
9	1.13	1.01
10	1.07	1.04
11	0.95	1.09
12	0.88	1.15

Use the log-transformed values as given in the dataset.

Use the following scatterplot to build your answers off of:

Part a

Here is the code for fitting the model:

summary(clot_mod)

Call:
lm(formula = LOGCONC ~ LOGDOSE, data = clot)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.097151 -0.026859 -0.003392  0.028279  0.095355 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  2.93620    0.04230   69.41 9.39e-15 ***
LOGDOSE     -1.78501    0.05267  -33.89 1.18e-11 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.05589 on 10 degrees of freedom
Multiple R-squared:  0.9914,    Adjusted R-squared:  0.9905 
F-statistic:  1149 on 1 and 10 DF,  p-value: 1.182e-11

Part b

term	df	sumsq	meansq	statistic	p.value
LOGDOSE	1	3.58845400	3.588454000	1148.759	1.182233e-11
Residuals	10	0.03123767	0.003123767	NA	NA

Part c

\[ F = 1148.759 \]

Part d

Not shown bc answer is complete solution

Question 2

Part a

[1] -0.9956757

Part b

term	df	sumsq	meansq	statistic	p.value
LOGDOSE	1	3.58845400	3.588454000	1148.759	1.182233e-11
Residuals	10	0.03123767	0.003123767	NA	NA

[1] 0.9913701

Part c

Not given

Question 3

if(!require(Sleuth3)) { install.packages("Sleuth3"); library(Sleuth3) }
q1_data = ex0824 

Part a

Part b

Not given

Part c

Fit the regression model, display the regression table, and write out the fitted regression line.

term	estimate	std.error	statistic	p.value
(Intercept)	47.0521633	0.50421678	93.31733	0.000000e+00
Age	-0.6957134	0.02937509	-23.68378	1.168799e-88

\[\begin{aligned} \widehat{\text{RR}} &= 47.05 -0.7 \cdot \text{Age} \end{aligned}\]

Part d

[1] "Rate"       "Age"        ".fitted"    ".resid"     ".hat"      
[6] ".sigma"     ".cooksd"    ".std.resid"

`stat_bin()` using `bins = 30`. Pick better value `binwidth`.

Part e

Part f

statistic	p.value	method
0.9782974	6.177141e-08	Shapiro-Wilk normality test

Part g

Part h

# A tibble: 0 × 8
# ℹ 8 variables: Rate <int>, Age <dbl>, .fitted <dbl>, .resid <dbl>,
#   .hat <dbl>, .sigma <dbl>, .cooksd <dbl>, .std.resid <dbl>

Part i

# A tibble: 618 × 8
    Rate   Age .fitted .resid    .hat .sigma .cooksd .std.resid
   <int> <dbl>   <dbl>  <dbl>   <dbl>  <dbl>   <dbl>      <dbl>
 1    78   1.9    45.7   32.3 0.00347   7.74  0.0296       4.12
 2    75   0.5    46.7   28.3 0.00395   7.76  0.0259       3.62
 3    73   3.1    44.9   28.1 0.00310   7.77  0.0201       3.59
 4    72   1.8    45.8   26.2 0.00350   7.78  0.0197       3.35
 5    70   2.5    45.3   24.7 0.00328   7.78  0.0164       3.15
 6    69   1.9    45.7   23.3 0.00347   7.79  0.0154       2.97
 7    69   4.5    43.9   25.1 0.00273   7.78  0.0140       3.20
 8    66   4.9    43.6   22.4 0.00263   7.80  0.0107       2.85
 9    27   1.5    46.0  -19.0 0.00360   7.81  0.0107      -2.43
10    48  25.3    29.5   18.5 0.00361   7.81  0.0102       2.37
# ℹ 608 more rows

Part j

`stat_bin()` using `bins = 30`. Pick better value `binwidth`.

Part k

Warning: The dot-dot notation (`..density..`) was deprecated in ggplot2 3.4.0.
ℹ Please use `after_stat(density)` instead.
ℹ The deprecated feature was likely used in the describedata package.
  Please report the issue at
  <https://github.com/craigjmcgowan/describedata/issues>.

Part l

Not given

Part m

Part n

For the log-transformed Rate:

q1_model2 = lm (Rate_log ~ Age, data = q1_data2)
q1_mod_t2 = tidy(q1_model2)
q1_mod_t2 %>% gt()

term	estimate	std.error	statistic	p.value
(Intercept)	3.84511854	0.0126276539	304.49984	0.000000e+00
Age	-0.01900896	0.0007356727	-25.83888	2.740318e-100

Part o

For log-transformed Rate:

Part p

For log-transformed Rate:

Part q

Not given