Untitled
Hongyu Chen
Tuesday, October 28, 2014
Analysis of covariance
Hongyu Chen
RPI, RIN:661405156
Oct.27 V1.0
1. Setting
System under test
Below is the installation and initial examination of the dataset:
library("Ecdat")## Loading required package: Ecfun
##
## Attaching package: 'Ecdat'
##
## The following object is masked from 'package:datasets':
##
## Orangedata<-BudgetFood
head(data)## wfood totexp age size town sex
## 1 0.4677 1290941 43 5 2 man
## 2 0.3130 1277978 40 3 2 man
## 3 0.3765 845852 28 3 2 man
## 4 0.4397 527698 60 1 2 woman
## 5 0.4036 1103220 37 5 2 man
## 6 0.1993 1768128 35 4 2 mansummary(data)## wfood totexp age size
## Min. :0.000 Min. : 14601 Min. :16.0 Min. : 1.0
## 1st Qu.:0.258 1st Qu.: 449820 1st Qu.:38.0 1st Qu.: 2.0
## Median :0.364 Median : 731114 Median :50.0 Median : 4.0
## Mean :0.378 Mean : 865550 Mean :50.5 Mean : 3.7
## 3rd Qu.:0.485 3rd Qu.: 1112533 3rd Qu.:62.0 3rd Qu.: 5.0
## Max. :0.997 Max. :11397547 Max. :99.0 Max. :37.0
## town sex
## Min. :1.00 man :20624
## 1st Qu.:2.00 woman: 3347
## Median :4.00 NA's : 1
## Mean :3.24
## 3rd Qu.:4.00
## Max. :5.00Factors and Levels
In this study we focus on particular group of people whose total household expenditure is relatively small.
One factor that interested in is the size of town where the household is placed, it is categorised into 5 groups: 1 for small towns, 5 for big ones. There are two explanatory variables, the age of reference person of the household and total expenditure.
Below is selection of sample data.
# factors and levels setting
P<-data$totexp<=50000
sample<-data[P,]
sample## wfood totexp age size town sex
## 556 0.45383 39072 85 1 2 woman
## 1353 0.63494 47992 73 1 3 man
## 1696 0.35556 23400 77 1 1 woman
## 2093 0.39792 24176 66 2 3 man
## 2767 0.67126 46480 88 1 4 woman
## 4286 0.54171 48764 78 1 1 woman
## 4616 0.00000 22840 75 1 3 woman
## 4918 0.00000 46800 72 1 4 woman
## 5794 0.00000 14601 53 1 3 woman
## 6178 0.17825 29173 63 1 1 woman
## 6356 0.57179 49564 74 2 3 man
## 7333 0.48296 47052 82 1 1 woman
## 8090 0.53947 47424 78 2 2 man
## 8210 0.63381 30602 74 1 2 woman
## 8213 0.40000 29900 70 3 2 woman
## 9009 0.75801 34712 69 1 4 woman
## 9030 0.16663 20284 72 1 3 woman
## 9851 0.63925 28064 88 1 4 man
## 10535 0.05772 16216 84 2 2 woman
## 10569 0.37705 37788 67 2 1 woman
## 10704 0.44583 27176 73 1 4 man
## 11748 0.79393 44014 64 1 2 man
## 11789 0.77432 23236 78 1 4 woman
## 11960 0.70719 37868 82 1 1 woman
## 13540 0.23854 38584 76 1 4 woman
## 13564 0.57769 27184 86 1 4 woman
## 15041 0.61867 39336 73 1 2 woman
## 15105 0.08816 25364 71 1 4 woman
## 15347 0.49239 46256 67 2 4 woman
## 16382 0.00000 40560 65 3 2 man
## 17658 0.00000 22786 66 1 4 woman
## 19123 0.22862 41169 70 1 5 woman
## 20329 0.58266 40339 76 1 1 woman
## 20583 0.64191 45770 72 1 1 woman
## 22802 0.37539 41695 73 1 1 woman
## 23175 0.51520 47236 80 1 1 woman
## 23263 0.84146 42640 78 1 1 man
## 23869 0.80620 31476 77 1 4 womanstr(sample)## 'data.frame': 38 obs. of 6 variables:
## $ wfood : num 0.454 0.635 0.356 0.398 0.671 ...
## $ totexp: num 39072 47992 23400 24176 46480 ...
## $ age : num 85 73 77 66 88 78 75 72 53 63 ...
## $ size : num 1 1 1 2 1 1 1 1 1 1 ...
## $ town : num 2 3 1 3 4 1 3 4 3 1 ...
## $ sex : Factor w/ 2 levels "man","woman": 2 1 2 1 2 2 2 2 2 2 ...Continuous variables (if any)
There are several continous variables in this test. Percentage of food expenditure, total expenditure and age of reference person are all continous variables.
Response variables
Percentage of total expenditure spent on food of a household is assigned as responsible variable, which is ‘wfood’ in the data set.
The Data: How is it organized and what does it look like?
The dataframe contains columns about persentage of expenditure on food, total expenditure, age of reference person, size of household, size of town they live, and sex of reference person.
This analysis is interested in how size of town people live affects percentage of expenditure on food, there are two explanatory variables which are age and total expenditure.
Randomization
Survey was condected within 23972 households in Spain, from information provided, it is fair to say all factors including age, household size, town size are completely randomized.
2. (Experimental) Design
How will the experiment be organized and conducted to test the hypothesis?
We select one insterested factor ‘town’ and other two independent explanatory variables, ‘age’ and ‘totexp’. Then test whether the size of town people live in could influence the percentage of their expenditure on food, also we are going to analyze the covariance of explanatory variables.
Null hypothesis H0:Variance of the percentage of expenditure spent on food is only due to randomization.
Alternative hypothesis H1: variance of the percentage of expenditure spent on food is due to other than randomizaion.
What is the rationale for this design?
Engel’s law is an observation in economics stating that as income rises, the proportion of income spent on food falls. In other words, the income elasticity of demand of food is between 0 and 1. Engel’s coefficient is a significant parameter to evaluate quality of life.
Using covariance analysis, we can further investigate how a factor affect response variable with the help of explanatory variables.
Randomize: What is the Randomization Scheme?
Randomization depends on the way data collected which can be assumed as randomly gethered.
Replicate: Are there replicates and/or repeated measures?
There is no replicates or repeated measures in this test.
Block: Did you use blocking in the design?
We set a restriction on total expenditure. In this case we investigate households with relatively low expenditure as less than 50,000.
3. (Statistical) Analysis
(Exploratory Data Analysis) Graphics and descriptive summary
#Histogram plot
par(mfrow=c(1,3));for (i in c(1,2,3)) hist(sample[,i],main = names(sample)[i])
#Boxplot
boxplot(wfood~town,data=sample,xlab="size of town",ylab="percentage of expenditure on food")
The boxplot reveal how the size of town affect expenditure percentage on food, there are obvious in the mean of ‘wfood’ when size of town is different, which may indicate that variance of expenditure percentage spent on food is due to size of town.
#Plot each variable interested against others
plot(sample[,c(1,2,3,5)])
plot(wfood~age,pch=as.character(town),data=sample)
plot(wfood~totexp,pch=as.character(town),data=sample)
From plots about each variable against others, it seems when expenditure percentage spent on food is dependent variable, it is hard to tell if the relationship between dependent and independent variables is positve or negative correlation. Plot is widely distributed.
In the plot about ‘wfood’ ‘age’ and ‘wfood’ ‘totexp’, town size effect is also not clear.
Testing
ANCOVA:
# Analysis of covariance
sample$town=as.factor(sample$town)
sampleaov=lm(wfood~totexp+age+town, data=sample)
anova(sampleaov)## Analysis of Variance Table
##
## Response: wfood
## Df Sum Sq Mean Sq F value Pr(>F)
## totexp 1 0.434 0.434 7.69 0.0093 **
## age 1 0.273 0.273 4.84 0.0354 *
## town 4 0.061 0.015 0.27 0.8936
## Residuals 31 1.750 0.056
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1anova(lm(wfood~town,data=sample))## Analysis of Variance Table
##
## Response: wfood
## Df Sum Sq Mean Sq F value Pr(>F)
## town 4 0.226 0.0566 0.81 0.53
## Residuals 33 2.292 0.0695ANCOVA result as above reveals that P-value of both total expenditure and age of reference person are less than 0.05, which indicates variation of these two independent explanatory variables may cause variation of percentage of expenditure spent on food.
By contrast, the factor we interested in, namely the size of town, has a P-value much larger than 0.05. Therefore we could reject the null hypothesis and accept the alternative hypothesis that variation of percentage of expenditure spent on food is due to something other than randomization.
Estimation (of Parameters)
# Summary of linear model
sampleaov=lm(wfood~totexp+age+town, data=sample)
summary(sampleaov)##
## Call:
## lm(formula = wfood ~ totexp + age + town, data = sample)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.5422 -0.1201 0.0034 0.1268 0.3867
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -6.37e-01 4.38e-01 -1.45 0.157
## totexp 9.18e-06 4.13e-06 2.22 0.034 *
## age 1.03e-02 5.61e-03 1.84 0.076 .
## town2 -2.08e-02 1.12e-01 -0.19 0.853
## town3 -5.33e-02 1.31e-01 -0.41 0.687
## town4 6.03e-03 1.03e-01 0.06 0.954
## town5 -2.35e-01 2.50e-01 -0.94 0.354
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.238 on 31 degrees of freedom
## Multiple R-squared: 0.305, Adjusted R-squared: 0.171
## F-statistic: 2.27 on 6 and 31 DF, p-value: 0.0624confint(sampleaov)## 2.5 % 97.5 %
## (Intercept) -1.531e+00 0.2574579
## totexp 7.604e-07 0.0000176
## age -1.126e-03 0.0217705
## town2 -2.489e-01 0.2072363
## town3 -3.209e-01 0.2143248
## town4 -2.050e-01 0.2170481
## town5 -7.451e-01 0.2745875From estimates result, only total expenditure returns a P-value smaller than 0.05
Diagnostics/Model Adequacy Checking
#Q-Q norm
qqnorm(residuals(sampleaov))
qqline(residuals(sampleaov))
#Plot of fitted and residuals
plot(fitted(sampleaov),residuals(sampleaov))
Q-Q norm shows a linear pattern, also points are evenly distributed on each side of zero, indicating the model used before is valid.
#Total expenditure with different town size
plot(wfood~totexp,pch=unclass(town),data=sample)
for (i in 1:4) abline(lm(wfood~totexp,data=sample[sample$town==i,]))
#Age of referece person with different town size
plot(wfood~age,pch=unclass(town),data=sample)
for (i in 1:4) abline(lm(wfood~age,data=sample[sample$town==i,]))
None of plots above has parallel lines in terms of different size of town.
4. References to the literature
https://www2.bc.edu/~lewbel/CollectiveEngel.pdf https://en.wikipedia.org/wiki/Engel%27s_law