Background

Overview

This background is quoted directly from the article “A Fine is a Price”.

There are two types of day-care centers in Israel: private and public. A study was conducted in 10 private day-care centers in the city of Haifa from January to June 1998. All of these centers are located in the same part of town, and there is no important difference among them. During the day children are organized into groups according to age, from 1 to 4 years old. Each day-care center is allowed to hold a maximum of 35 children. In some exceptional cases a few additional children are allowed. The fee for each child is NIS 1,400 per month. (The NIS is the New Israeli Shekel.) At the time of the study, a U.S. dollar was worth approximately NIS 3.68, so the fee was about $380 at that time.

The contract signed at the beginning of the year states that the day-care center operates between 0730 and 1600. There is no mention of what happens if parents come late to pick up their children. In particular, before the beginning of the study, there was no fine for coming late. When parents did not come on time, one of the teachers had to wait with the children concerned. Teachers would rotate in this task, which is considered part of the job of a teacher, a fact that is clearly explained when a teacher is hired. Parents rarely came after 1630.

A natural option [to fix the problem of parents showing up late] is to introduce a fine: every time a parent comes late, [they] will have to pay a fine. Will that reduce the number of parents who come late? If the fine is removed, will things revert back to the way they were originally?

The overall period of the study was 20 weeks. In the first 4 weeks we simply recorded the number of parents who arrived late each week. At the beginning of the fifth week, we introduced a fine in six of the 10 day-care centers, which had been selected randomly. The announcement of the fine was made with a note posted on the bulletin board of the day-care center. Parents tend to look at this board every day, since important announcements are posted there. The announcement specified that the fine would be NIS 10 for a delay of 10 minutes or more. The fine was per child; thus, if parents had two children in the center and they came late, they had to pay NIS 20. Payment was made to the principal of the day-care center at the end of the month. Since monthly payments are made to the owner during the year, the fines were added to those amounts. The money was paid to the owner, rather then to the teacher who was staying late (and did not get any additional money). The teachers were informed of the fine but not of the study. Registering the names of parents who came late was a common practice in any case.

At the beginning of the seventeenth week, the fine was removed with no explanation. Notice of the cancellation was posted on the board. If parents asked why the fines were removed, the principals were instructed to reply that the fine had been a trial for a limited time and that the results of this trial were now being evaluated.

A comparison with other fines in Israel may give an idea of the size of the penalty that was introduced. A fine of NIS 10 is relatively small but not insignificant. In comparison, the fine for illegal parking is NIS 75; the fine for driving through a red light is NIS 1,000 plus penalties; the fine for not collecting the droppings of a dog is NIS 360. For many of these violations, however, detection and enforcement are low or, as in the case of dog dirt, nonexistent in practice. A baby-sitter earns between NIS 15 and NIS 20 per hour. The average gross salary per month in Israel at the time of the study was NIS 5,595.

Hypothesis

The test done is a Two Way ANOVA

  1. Does the type of Week affect the average number of late children? Factor: Week with levels \(A\) and \(B\). \[ H_0: \mu_A = \mu_B = \mu \] \[ H_a: \mu_A \neq \mu_B \]

  2. Does the Treatment affect the average number of late children? Factor: Treatment with levels \(L\), \(M\), and \(H\). \[ H_0: \mu_L = \mu_M = \mu_H = \mu \] \[ H_a: \mu_i \neq \mu \ \text{for at least one}\ i\in\{1=L,2=M,3=H\} \]

  3. Does the effect of Treatment change for different types of Week? (Does the effect of Week change for different levels of Treatment?) In other words, is there an interaction between Week and Treatment?

\[ H_0: \text{The effect of Treatment is the same for all Weeks.} \] \[ H_a: \text{The effect of Treatment is not the same for all types of Weeks.} \]

A significance level of \(\alpha = 0.05\) will be used for this study.

The Data (Wide)

The late Day Care Center data is shown here in the “wide data format”.

#Show the full width of the "Wide" version of the late data:
pander(late, split.tables = Inf)
Treatment Center No.ofChidren Week1 Week2 Week3 Week4 Week5 Week6 Week7 Week8 Week9 Week10 Week11 Week12 Week13 Week14 Week15 Week16 Week17 Week18 Week19 Week20
Fine 1 37 8 8 7 6 8 9 9 12 13 13 15 13 14 16 14 15 16 13 15 17
Fine 2 35 6 7 3 5 2 11 14 9 16 12 10 14 14 16 12 17 14 10 14 15
Fine 3 35 8 9 8 9 3 5 15 18 16 14 20 18 25 22 27 19 20 23 23 22
Fine 4 34 10 3 14 9 6 24 8 22 22 19 25 18 23 22 24 17 15 23 25 18
Fine 5 33 13 12 9 13 15 10 27 28 35 10 24 32 29 29 26 31 26 35 29 28
Fine 6 28 5 8 7 5 5 9 12 14 19 17 14 13 10 15 14 16 6 12 17 13
Control 7 35 7 10 12 6 4 13 7 8 5 12 3 5 6 13 7 4 7 10 4 6
Control 8 34 12 9 14 18 10 11 6 15 14 13 7 12 9 9 17 8 5 11 8 13
Control 9 34 3 4 9 3 3 5 9 5 2 7 6 6 9 4 9 2 3 8 3 5
Control 10 32 15 13 13 12 10 9 15 15 15 10 17 12 13 11 14 17 12 9 15 13

The Data (Long)

The Late Day Care Center data is shown here in the “long data format”.

pander(Late)
Treatment Center No.ofChidren Week NumberofLateChildren Quarter
Fine 1 37 1 8 First
Fine 1 37 2 8 First
Fine 1 37 3 7 First
Fine 1 37 4 6 First
Fine 1 37 5 8 First
Fine 1 37 6 9 Second
Fine 1 37 7 9 Second
Fine 1 37 8 12 Second
Fine 1 37 9 13 Second
Fine 1 37 10 13 Second
Fine 1 37 11 15 Third
Fine 1 37 12 13 Third
Fine 1 37 13 14 Third
Fine 1 37 14 16 Third
Fine 1 37 15 14 Third
Fine 1 37 16 15 Forth
Fine 1 37 17 16 Forth
Fine 1 37 18 13 Forth
Fine 1 37 19 15 Forth
Fine 1 37 20 17 Forth
Fine 2 35 1 6 First
Fine 2 35 2 7 First
Fine 2 35 3 3 First
Fine 2 35 4 5 First
Fine 2 35 5 2 First
Fine 2 35 6 11 Second
Fine 2 35 7 14 Second
Fine 2 35 8 9 Second
Fine 2 35 9 16 Second
Fine 2 35 10 12 Second
Fine 2 35 11 10 Third
Fine 2 35 12 14 Third
Fine 2 35 13 14 Third
Fine 2 35 14 16 Third
Fine 2 35 15 12 Third
Fine 2 35 16 17 Forth
Fine 2 35 17 14 Forth
Fine 2 35 18 10 Forth
Fine 2 35 19 14 Forth
Fine 2 35 20 15 Forth
Fine 3 35 1 8 First
Fine 3 35 2 9 First
Fine 3 35 3 8 First
Fine 3 35 4 9 First
Fine 3 35 5 3 First
Fine 3 35 6 5 Second
Fine 3 35 7 15 Second
Fine 3 35 8 18 Second
Fine 3 35 9 16 Second
Fine 3 35 10 14 Second
Fine 3 35 11 20 Third
Fine 3 35 12 18 Third
Fine 3 35 13 25 Third
Fine 3 35 14 22 Third
Fine 3 35 15 27 Third
Fine 3 35 16 19 Forth
Fine 3 35 17 20 Forth
Fine 3 35 18 23 Forth
Fine 3 35 19 23 Forth
Fine 3 35 20 22 Forth
Fine 4 34 1 10 First
Fine 4 34 2 3 First
Fine 4 34 3 14 First
Fine 4 34 4 9 First
Fine 4 34 5 6 First
Fine 4 34 6 24 Second
Fine 4 34 7 8 Second
Fine 4 34 8 22 Second
Fine 4 34 9 22 Second
Fine 4 34 10 19 Second
Fine 4 34 11 25 Third
Fine 4 34 12 18 Third
Fine 4 34 13 23 Third
Fine 4 34 14 22 Third
Fine 4 34 15 24 Third
Fine 4 34 16 17 Forth
Fine 4 34 17 15 Forth
Fine 4 34 18 23 Forth
Fine 4 34 19 25 Forth
Fine 4 34 20 18 Forth
Fine 5 33 1 13 First
Fine 5 33 2 12 First
Fine 5 33 3 9 First
Fine 5 33 4 13 First
Fine 5 33 5 15 First
Fine 5 33 6 10 Second
Fine 5 33 7 27 Second
Fine 5 33 8 28 Second
Fine 5 33 9 35 Second
Fine 5 33 10 10 Second
Fine 5 33 11 24 Third
Fine 5 33 12 32 Third
Fine 5 33 13 29 Third
Fine 5 33 14 29 Third
Fine 5 33 15 26 Third
Fine 5 33 16 31 Forth
Fine 5 33 17 26 Forth
Fine 5 33 18 35 Forth
Fine 5 33 19 29 Forth
Fine 5 33 20 28 Forth
Fine 6 28 1 5 First
Fine 6 28 2 8 First
Fine 6 28 3 7 First
Fine 6 28 4 5 First
Fine 6 28 5 5 First
Fine 6 28 6 9 Second
Fine 6 28 7 12 Second
Fine 6 28 8 14 Second
Fine 6 28 9 19 Second
Fine 6 28 10 17 Second
Fine 6 28 11 14 Third
Fine 6 28 12 13 Third
Fine 6 28 13 10 Third
Fine 6 28 14 15 Third
Fine 6 28 15 14 Third
Fine 6 28 16 16 Forth
Fine 6 28 17 6 Forth
Fine 6 28 18 12 Forth
Fine 6 28 19 17 Forth
Fine 6 28 20 13 Forth
Control 7 35 1 7 First
Control 7 35 2 10 First
Control 7 35 3 12 First
Control 7 35 4 6 First
Control 7 35 5 4 First
Control 7 35 6 13 Second
Control 7 35 7 7 Second
Control 7 35 8 8 Second
Control 7 35 9 5 Second
Control 7 35 10 12 Second
Control 7 35 11 3 Third
Control 7 35 12 5 Third
Control 7 35 13 6 Third
Control 7 35 14 13 Third
Control 7 35 15 7 Third
Control 7 35 16 4 Forth
Control 7 35 17 7 Forth
Control 7 35 18 10 Forth
Control 7 35 19 4 Forth
Control 7 35 20 6 Forth
Control 8 34 1 12 First
Control 8 34 2 9 First
Control 8 34 3 14 First
Control 8 34 4 18 First
Control 8 34 5 10 First
Control 8 34 6 11 Second
Control 8 34 7 6 Second
Control 8 34 8 15 Second
Control 8 34 9 14 Second
Control 8 34 10 13 Second
Control 8 34 11 7 Third
Control 8 34 12 12 Third
Control 8 34 13 9 Third
Control 8 34 14 9 Third
Control 8 34 15 17 Third
Control 8 34 16 8 Forth
Control 8 34 17 5 Forth
Control 8 34 18 11 Forth
Control 8 34 19 8 Forth
Control 8 34 20 13 Forth
Control 9 34 1 3 First
Control 9 34 2 4 First
Control 9 34 3 9 First
Control 9 34 4 3 First
Control 9 34 5 3 First
Control 9 34 6 5 Second
Control 9 34 7 9 Second
Control 9 34 8 5 Second
Control 9 34 9 2 Second
Control 9 34 10 7 Second
Control 9 34 11 6 Third
Control 9 34 12 6 Third
Control 9 34 13 9 Third
Control 9 34 14 4 Third
Control 9 34 15 9 Third
Control 9 34 16 2 Forth
Control 9 34 17 3 Forth
Control 9 34 18 8 Forth
Control 9 34 19 3 Forth
Control 9 34 20 5 Forth
Control 10 32 1 15 First
Control 10 32 2 13 First
Control 10 32 3 13 First
Control 10 32 4 12 First
Control 10 32 5 10 First
Control 10 32 6 9 Second
Control 10 32 7 15 Second
Control 10 32 8 15 Second
Control 10 32 9 15 Second
Control 10 32 10 10 Second
Control 10 32 11 17 Third
Control 10 32 12 12 Third
Control 10 32 13 13 Third
Control 10 32 14 11 Third
Control 10 32 15 14 Third
Control 10 32 16 17 Forth
Control 10 32 17 12 Forth
Control 10 32 18 9 Forth
Control 10 32 19 15 Forth
Control 10 32 20 13 Forth

Data Analysis

Basic Statistics

StatsLate <- favstats(NumberofLateChildren ~ Week, data = Late)

pander(StatsLate[,-10])
Week min Q1 median Q3 max mean sd n
1 3 6.25 8 11.5 15 8.7 3.773 10
2 3 7.25 8.5 9.75 13 8.3 3.129 10
3 3 7.25 9 12.75 14 9.6 3.596 10
4 3 5.25 7.5 11.25 18 8.6 4.6 10
5 2 3.25 5.5 9.5 15 6.6 4.115 10
6 5 9 9.5 11 24 10.6 5.337 10
7 6 8.25 10.5 14.75 27 12.2 6.161 10
8 5 9.75 14.5 17.25 28 14.6 6.835 10
9 2 13.25 15.5 18.25 35 15.7 9.044 10
10 7 10.5 12.5 13.75 19 12.7 3.466 10
11 3 7.75 14.5 19.25 25 14.1 7.578 10
12 5 12 13 17 32 14.3 7.528 10
13 6 9.25 13.5 20.75 29 15.2 7.772 10
14 4 11.5 15.5 20.5 29 15.7 7.212 10
15 7 12.5 14 22.25 27 16.4 7.011 10
16 2 9.75 16.5 17 31 14.6 8.316 10
17 3 6.25 13 15.75 26 12.4 7.291 10
18 8 10 11.5 20.5 35 15.4 8.758 10
19 3 9.5 15 21.5 29 15.3 8.68 10
20 5 13 14 17.75 28 15 6.864 10

ANOVA Test

myaov <- aov(NumberofLateChildren ~ Quarter+Treatment+Quarter:Treatment, data=Late)

pander(myaov)
Analysis of Variance Model
  Df Sum Sq Mean Sq F value Pr(>F)
Quarter 3 1417 472.4 16.89 8.916e-10
Treatment 1 1740 1740 62.2 2.288e-13
Quarter:Treatment 3 1109 369.7 13.22 7.007e-08
Residuals 192 5371 27.98 NA NA 
par(mfrow = c(1,2))
plot(myaov, which=1:2)

Graphical Summaries

Quarters

ggplot(Late) + 
  geom_point(aes(x=Quarter, y=NumberofLateChildren)) +
  ggtitle("Significance of Quarter")

Treatment

ggplot(Late) + 
  geom_point(aes(x=Treatment, y=NumberofLateChildren)) +
  ggtitle("Significance of Treatment")

Treatment Depending on Quarter

xyplot(NumberofLateChildren ~ Quarter, data=Late, groups=Treatment, type=c("p","a"), main="Significance of the Interaction", auto.key=list(corner=c(1,1)))

Interpretation

The graphs above show the interaction of late children, treatment and quarter. The first quarter has the least about of late. Also the control treatment has the least amount of late children. The control treatment decreases the lateness of children over the quarters where as the fine treatment increases the lateness of children over the quarters.