Advanced Biostatistics HW 2

26 BE 7023 & 26 PH 7023: Advanced Biostatistics Autumn, 2017. MB Rao

Homework Sheet No. 2

1. Do a linear regression model with longevity as response variable and thorax and activity as predictors. Comment on the output.

#install.packages("faraway")
library(faraway)
#data(package="faraway")
data("fruitfly")
#head(fruitfly)
modelmultivar<-lm(longevity~activity+thorax, data = fruitfly)
summary(modelmultivar)

## 
## Call:
## lm(formula = longevity ~ activity + thorax, data = fruitfly)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -26.108  -7.014  -1.101   6.234  30.265 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   -48.749     10.850  -4.493 1.65e-05 ***
## activityone     2.637      2.984   0.884   0.3786    
## activitylow    -7.015      2.981  -2.353   0.0203 *  
## activitymany    4.139      3.027   1.367   0.1741    
## activityhigh  -20.004      3.016  -6.632 1.05e-09 ***
## thorax        134.341     12.731  10.552  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10.54 on 118 degrees of freedom
## Multiple R-squared:  0.6527, Adjusted R-squared:  0.638 
## F-statistic: 44.36 on 5 and 118 DF,  p-value: < 2.2e-16

Comments: 65% of the variability in longevity of the fruitfly can be explained by the length of the thorax and by extent of sexual activity.

Thorax length has significant influence on the longevity. Compared to fruitflies in isolation those with low or high activity have signficantly less longevity

2. We have fitted a simple linear regression model with longevity as response variable and activity as predictor in the class. Compare this model with the model in Question 1.

modelunivar<-lm(longevity~activity, data=fruitfly)
summary(modelunivar)

## 
## Call:
## lm(formula = longevity ~ activity, data = fruitfly)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -35.760  -8.760   0.329  11.200  32.440 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   63.5600     2.9263  21.720  < 2e-16 ***
## activityone    1.2400     4.1384   0.300    0.765    
## activitylow   -6.8000     4.1384  -1.643    0.103    
## activitymany   0.9817     4.1813   0.235    0.815    
## activityhigh -24.8400     4.1384  -6.002 2.16e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 14.63 on 119 degrees of freedom
## Multiple R-squared:  0.3251, Adjusted R-squared:  0.3024 
## F-statistic: 14.33 on 4 and 119 DF,  p-value: 1.411e-09

Comments: Sexual activity in fruitflies explain only about 32% of the variability in their longevity. Therefore, activity explains only a lower estimate of the actual variability in the longevity of the fruitflies. Additional factors such as length of the thorax make the model more robust

3. There is another data set ‘Animals’ similar to the ‘mammals’ data in spirit. The focus now is on these data. You are responsible for a thorough analysis of these data. Get the following items aired out completely.

library(MASS)
data("Animals")
head(Animals,7)

##                     body  brain
## Mountain beaver     1.35    8.1
## Cow               465.00  423.0
## Grey wolf          36.33  119.5
## Goat               27.66  115.0
## Guinea pig          1.04    5.5
## Dipliodocus     11700.00   50.0
## Asian elephant   2547.00 4603.0

dim(Animals)

## [1] 28  2

Comments Loading the package MASS and the dataset Animals, printing first six rows of Animals and determining the size of data

a. Scatter plot of log(body) vs log(brain)

plot(log(Animals$body), log(Animals$brain), xlab="Log(Body Weight)", ylab="Log(Brain Weight)", main="Plot of Body Wt. to Brain Wt.")

###b. Identification of some animals.

plot(log(Animals$body), log(Animals$brain), xlab="Log(Body Weight)", ylab="Log(Brain Weight)", main="Plot of Body Wt. to Brain Wt.")
identify(log(Animals$body), log(Animals$brain), labels=row.names(Animals))

## integer(0)

c. Fit a simple linear regression model: log(brain) ~ log(body). Comment on the output.

anim_reg<-lm(log(brain)~log(body), data=Animals)
summary(anim_reg)

## 
## Call:
## lm(formula = log(brain) ~ log(body), data = Animals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.2890 -0.6763  0.3316  0.8646  2.5835 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.55490    0.41314   6.184 1.53e-06 ***
## log(body)    0.49599    0.07817   6.345 1.02e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.532 on 26 degrees of freedom
## Multiple R-squared:  0.6076, Adjusted R-squared:  0.5925 
## F-statistic: 40.26 on 1 and 26 DF,  p-value: 1.017e-06

Comments Log(body weight) explains about 60% of the variability in the Log(brain weight) at a chance of Type 1 error very close to zero (p<0.0001)

d. Draw the line on the scatter plot.

plot(log(Animals$body), log(Animals$brain), xlab="Log(Body Weight)", ylab="Log(Brain Weight)", main="Plot of Body Wt. to Brain Wt.")
abline(anim_reg, lwd=2, col="red")

e. Comment what the dinosaurs are doing to the line.

Comment:The ratio of brain weight to body weight of dinosaurs are relatively lower compared to other mammals. Thus the points on the plot for the dinosaurs are further to the right and is decreasing the slope of the fitted line.

f. Remove the dinosaurs.

subsetAnimals<-subset(Animals, !(row.names(Animals)== "Dipliodocus" | 
    row.names(Animals)=="Triceratops" | row.names(Animals)== "Brachiosaurus"))
dim(subsetAnimals)

## [1] 25  2

head(subsetAnimals, 15)

##                     body  brain
## Mountain beaver     1.35    8.1
## Cow               465.00  423.0
## Grey wolf          36.33  119.5
## Goat               27.66  115.0
## Guinea pig          1.04    5.5
## Asian elephant   2547.00 4603.0
## Donkey            187.10  419.0
## Horse             521.00  655.0
## Potar monkey       10.00  115.0
## Cat                 3.30   25.6
## Giraffe           529.00  680.0
## Gorilla           207.00  406.0
## Human              62.00 1320.0
## African elephant 6654.00 5712.0
## Rhesus monkey       6.80  179.0

Comment The subset after removal of the three dinosaur species has 25 rows and 2 columns. Top 15 rows are shown above.

g. Replot the data after the logarithmic transformation.

plot(log(subsetAnimals$body), log(subsetAnimals$brain), 
    xlab="Log(Body Weight)", ylab="Log(Brain Weight)", 
    main="Body Wt. vs Brain Wt.:Data subset", cex.main=.8)

####Comment The plot of Log(Body Weight) vs. Log(Brain Weight) after removal of rows for Dinosaurs is shown above.

h. Identify some animals.

plot(log(subsetAnimals$body), log(subsetAnimals$brain), 
    xlab="Log(Body Weight)", ylab="Log(Brain Weight)", 
    main="Body Wt. vs Brain Wt.:Data subset", cex.main=.8)
identify(log(subsetAnimals$body), log(subsetAnimals$brain), labels=row.names(subsetAnimals))

## integer(0)

i. Fit the relevant regression model. Comment on the output.

subanim_reg<-lm(log(brain)~log(body), data=subsetAnimals)
summary(subanim_reg)

## 
## Call:
## lm(formula = log(brain) ~ log(body), data = subsetAnimals)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -0.9125 -0.4752 -0.1557  0.1940  1.9303 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.15041    0.20060   10.72 2.03e-10 ***
## log(body)    0.75226    0.04572   16.45 3.24e-14 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.7258 on 23 degrees of freedom
## Multiple R-squared:  0.9217, Adjusted R-squared:  0.9183 
## F-statistic: 270.7 on 1 and 23 DF,  p-value: 3.243e-14

Comment Log(Body Weight) explains nearly 92% of the variability in the Log(Brain Weight) after removal of the dinosaurs data points. Thus, the model performance is improved in the data subset.

j. Draw the line on the scatter plot.

plot(log(subsetAnimals$body), log(subsetAnimals$brain), 
    xlab="Log(Body Weight)", ylab="Log(Brain Weight)", 
    main="Body Wt. vs Brain Wt. in Data subset after removal of Dinosaurs", cex.main=.8)
abline(subanim_reg, lwd=2, col="blue")

k. Rewrite the model in terms of the original variables.

Log(brain) =2.15041 + 0.75226 x log(body)

=log (8.588379) + 0.75226 x log(body)

=log (8.588379) + log(body)0.75226

=log (8.588379 x body0.75226)

Brain/Body^0.75226 =8.588379

l. Compare this model with the model coming from the ‘mammals.’

#data(package="MASS")
data(mammals)
#head(mammals)
mamm_reg<-lm(log(mammals$brain)~log(mammals$body))
summary(mamm_reg)

## 
## Call:
## lm(formula = log(mammals$brain) ~ log(mammals$body))
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.71550 -0.49228 -0.06162  0.43597  1.94829 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)        2.13479    0.09604   22.23   <2e-16 ***
## log(mammals$body)  0.75169    0.02846   26.41   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6943 on 60 degrees of freedom
## Multiple R-squared:  0.9208, Adjusted R-squared:  0.9195 
## F-statistic: 697.4 on 1 and 60 DF,  p-value: < 2.2e-16

Comments About 92% of the variability in the brain weight of the mammals can be explained by the differences in the body weight (probability of error close to zero). This is very similar to the output from animals subset data after removal of dinosaurs

Expression of the model in terms of original variables

Log(brain) =2.13479 + 0.75169 × Log (Brain)

=Log (8.455271) + Log (Brain)^0.75169

=Log (8.455271 × Brain^0.75169)

Thus, for the mammals data, Brain/Body^(0.75169) = 8.46

For the animals subset data (excluding dinosaurs), Brain/Body^(0.75226) = 8.59

For the animals original data (including dinosaurs),Brain/Body^(0.495599) = 12.87

Thus, the ratios of the brain wt. : body wt. is very similar for mammals data and animals data (excluding dinosaurs).

m. Calculate the ratio of the model for each animal in the data. Arrange the ratios in increasing order of magnitude. Comment on the output.

ratio=(subsetAnimals$brain)/((subsetAnimals$body)^(0.75226))
ratio

##  [1]  6.463089  4.166028  8.010187  9.463464  5.340098 12.613094  8.185096
##  [8]  5.922062 20.344070 10.427559  6.078020  7.350454 59.187413  7.600391
## [15] 42.324309  3.860534  4.928277  6.830722  6.073388  8.528570  4.913372
## [22] 22.468201  4.950334 14.602130  3.448548

subsetAnimals1<-cbind(subsetAnimals, ratio)
subsetAnimals1<-subsetAnimals1[order(ratio),]
head(subsetAnimals1)

##                  body brain    ratio
## Pig            192.00 180.0 3.448548
## Kangaroo        35.00  56.0 3.860534
## Cow            465.00 423.0 4.166028
## Jaguar         100.00 157.0 4.913372
## Golden hamster   0.12   1.0 4.928277
## Rat              0.28   1.9 4.950334

tail(subsetAnimals1)

##                    body brain    ratio
## Asian elephant 2547.000  4603 12.61309
## Mole              0.122     3 14.60213
## Potar monkey     10.000   115 20.34407
## Chimpanzee       52.160   440 22.46820
## Rhesus monkey     6.800   179 42.32431
## Human            62.000  1320 59.18741

subsetAnimals1

##                      body  brain     ratio
## Pig               192.000  180.0  3.448548
## Kangaroo           35.000   56.0  3.860534
## Cow               465.000  423.0  4.166028
## Jaguar            100.000  157.0  4.913372
## Golden hamster      0.120    1.0  4.928277
## Rat                 0.280    1.9  4.950334
## Guinea pig          1.040    5.5  5.340098
## Horse             521.000  655.0  5.922062
## Rabbit              2.500   12.1  6.073388
## Giraffe           529.000  680.0  6.078020
## Mountain beaver     1.350    8.1  6.463089
## Mouse               0.023    0.4  6.830722
## Gorilla           207.000  406.0  7.350454
## African elephant 6654.000 5712.0  7.600391
## Grey wolf          36.330  119.5  8.010187
## Donkey            187.100  419.0  8.185096
## Sheep              55.500  175.0  8.528570
## Goat               27.660  115.0  9.463464
## Cat                 3.300   25.6 10.427559
## Asian elephant   2547.000 4603.0 12.613094
## Mole                0.122    3.0 14.602130
## Potar monkey       10.000  115.0 20.344070
## Chimpanzee         52.160  440.0 22.468201
## Rhesus monkey       6.800  179.0 42.324309
## Human              62.000 1320.0 59.187413

CommentPigs, Kangaroo, Cows are some of the animals with lowest ratio, whereas Humans and other primates have highest ratio of Brain/Body wts.

n. Are humans outliers?

#install.packages("car")
library(car)

## 
## Attaching package: 'car'

## The following objects are masked from 'package:faraway':
## 
##     logit, vif

outlierTest(anim_reg)

## 
## No Studentized residuals with Bonferonni p < 0.05
## Largest |rstudent|:
##              rstudent unadjusted p-value Bonferonni p
## Dipliodocus -2.507366           0.019026      0.53272

outlierTest(subanim_reg)

## 
## No Studentized residuals with Bonferonni p < 0.05
## Largest |rstudent|:
##       rstudent unadjusted p-value Bonferonni p
## Human 3.231759          0.0038339     0.095847

Comments The dinosaur species Dipliodocus is an outlier in the original Animal dataset. After removal of all the dinosaurs, the humans are outlier in the remaining subset.

Advanced Biostatistics HW 2

Umesh Singh

September 18, 2017

26 BE 7023 & 26 PH 7023: Advanced Biostatistics Autumn, 2017. MB Rao

1. Do a linear regression model with longevity as response variable and thorax and activity as predictors. Comment on the output.

Comments: 65% of the variability in longevity of the fruitfly can be explained by the length of the thorax and by extent of sexual activity.

Thorax length has significant influence on the longevity. Compared to fruitflies in isolation those with low or high activity have signficantly less longevity

2. We have fitted a simple linear regression model with longevity as response variable and activity as predictor in the class. Compare this model with the model in Question 1.

Comments: Sexual activity in fruitflies explain only about 32% of the variability in their longevity. Therefore, activity explains only a lower estimate of the actual variability in the longevity of the fruitflies. Additional factors such as length of the thorax make the model more robust

3. There is another data set ‘Animals’ similar to the ‘mammals’ data in spirit. The focus now is on these data. You are responsible for a thorough analysis of these data. Get the following items aired out completely.

Comments Loading the package MASS and the dataset Animals, printing first six rows of Animals and determining the size of data

a. Scatter plot of log(body) vs log(brain)

c. Fit a simple linear regression model: log(brain) ~ log(body). Comment on the output.

Comments Log(body weight) explains about 60% of the variability in the Log(brain weight) at a chance of Type 1 error very close to zero (p<0.0001)

d. Draw the line on the scatter plot.

e. Comment what the dinosaurs are doing to the line.

Comment:The ratio of brain weight to body weight of dinosaurs are relatively lower compared to other mammals. Thus the points on the plot for the dinosaurs are further to the right and is decreasing the slope of the fitted line.

f. Remove the dinosaurs.

Comment The subset after removal of the three dinosaur species has 25 rows and 2 columns. Top 15 rows are shown above.

g. Replot the data after the logarithmic transformation.

h. Identify some animals.

i. Fit the relevant regression model. Comment on the output.

Comment Log(Body Weight) explains nearly 92% of the variability in the Log(Brain Weight) after removal of the dinosaurs data points. Thus, the model performance is improved in the data subset.

j. Draw the line on the scatter plot.

k. Rewrite the model in terms of the original variables.

Log(brain) =2.15041 + 0.75226 x log(body)

=log (8.588379) + 0.75226 x log(body)

=log (8.588379) + log(body)0.75226

=log (8.588379 x body0.75226)

Brain/Body^0.75226 =8.588379

l. Compare this model with the model coming from the ‘mammals.’

Comments About 92% of the variability in the brain weight of the mammals can be explained by the differences in the body weight (probability of error close to zero). This is very similar to the output from animals subset data after removal of dinosaurs

Expression of the model in terms of original variables

Log(brain) =2.13479 + 0.75169 × Log (Brain)

=Log (8.455271) + Log (Brain)^0.75169

=Log (8.455271 × Brain^0.75169)

Thus, for the mammals data, Brain/Body^(0.75169) = 8.46

For the animals subset data (excluding dinosaurs), Brain/Body^(0.75226) = 8.59

For the animals original data (including dinosaurs),Brain/Body^(0.495599) = 12.87

Thus, the ratios of the brain wt. : body wt. is very similar for mammals data and animals data (excluding dinosaurs).

m. Calculate the ratio of the model for each animal in the data. Arrange the ratios in increasing order of magnitude. Comment on the output.

CommentPigs, Kangaroo, Cows are some of the animals with lowest ratio, whereas Humans and other primates have highest ratio of Brain/Body wts.

n. Are humans outliers?

Comments The dinosaur species Dipliodocus is an outlier in the original Animal dataset. After removal of all the dinosaurs, the humans are outlier in the remaining subset.