Recipe 1: One Factor, Two Level Experiments

Recipes for the Design of Experiments

Zoe Konrad

Rensselaer Polytechnic Institute

Fall 2014 v1

1. Setting

System under test

The dataset contains basic information about every flight departing out of NYC airports in 2013.

library("nycflights13")
x<-flights
attach(x)
head(x)

##   year month day dep_time dep_delay arr_time arr_delay carrier tailnum
## 1 2013     1   1      517         2      830        11      UA  N14228
## 2 2013     1   1      533         4      850        20      UA  N24211
## 3 2013     1   1      542         2      923        33      AA  N619AA
## 4 2013     1   1      544        -1     1004       -18      B6  N804JB
## 5 2013     1   1      554        -6      812       -25      DL  N668DN
## 6 2013     1   1      554        -4      740        12      UA  N39463
##   flight origin dest air_time distance hour minute
## 1   1545    EWR  IAH      227     1400    5     17
## 2   1714    LGA  IAH      227     1416    5     33
## 3   1141    JFK  MIA      160     1089    5     42
## 4    725    JFK  BQN      183     1576    5     44
## 5    461    LGA  ATL      116      762    5     54
## 6   1696    EWR  ORD      150      719    5     54

The hypothesis under test is that Newark and JFK have a non-zero difference in mean departure delay times.

Factors and Levels

The factor we are intersted in for this analysis is origin, the airport of departure. We will look at two levels: JFK and EWR.

summary(as.factor(origin))

##    EWR    JFK    LGA 
## 120835 111279 104662

Response variables

The response variable is the departure delay, in minutes.

summary(dep_delay)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##     -43      -5      -2      13      11    1300    8255

Note that negative delay signifies an early departure.

2. (Experimental) Design

The dataset contains every flight departing out of NYC airports in 2013, thus the ‘experiment’ is fully complete and randomized. Every flight out of each airport could be viewed as a repeated measure in the experiment. We use an independent 2-group t-test to compare the departure delays out of JFK and EWR.

3. (Statistical) Analysis

Exploratory Data Analysis Graphics

boxplot(dep_delay~origin, outline=FALSE)

plot of chunk unnamed-chunk-4

There does not appear to be a significant difference in means accross airports of origin.

Testing

EWR <- subset(x, origin =='EWR')
JFK <- subset(x, origin =='JFK')
t.test(EWR$dep_delay, JFK$dep_delay)

## 
##  Welch Two Sample t-test
## 
## data:  EWR$dep_delay and JFK$dep_delay
## t = 17.76, df = 226958, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  2.665 3.326
## sample estimates:
## mean of x mean of y 
##     15.11     12.11

The t-test concludes that there is a statistically significant difference in means between EWR and JFK. Thus, we reject the null hypothesis and conclude that randomization alone cannot account for the variation in departure delay times.

Estimation (of Parameters)

summary(EWR$dep_delay)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##     -25      -4      -1      15      15    1130    3239

summary(JFK$dep_delay)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    NA's 
##   -43.0    -5.0    -1.0    12.1    10.0  1300.0    1863

Flights our of EWR have a mean departure delay of 15.11 minutes where ase flights out of JFK have a mean departure delay of 12.11 minutes. With 95% confidence from the t-test above, flights out of EWR will be 2.67 to 3.33 minutes more delayed in their departure than flights out of JFK.

Diagnostics/Model Adequacy Checking

The response variable departure delay violates the assumption of normality, but, the t-test is described as a robust test with respect to the assumption of normality, so we do not necessarily disregard the model.

qqnorm(dep_delay)
qqline(dep_delay)

plot of chunk unnamed-chunk-7