Week 5 Analysis

Background

Overview

An experiment was performed in Israel where a fine was introduced for parents arriving late to pick up their children from day-care. There was a control group of day-care centers that did not implement the fine during the entire period. There was a treatment group where there was no fine for the first four weeks, then a fine for twelve weeks and then removed the fine for four weeks.

In this study, we will compare the various relationships between the control group and the fined group and the different periods of the study concerning the number of children that are picked up late. We will be using the following questions and hypotheses.

Does implementing a fine effect the average number of children picked up late? \[H_0: \mu_{Control} = \mu_{Fined} = \mu\] \[H_a: \mu_i \neq \mu \ \text{for at least one}\ i\in\{1 = Control,\ 2 = Fined\}\]

Is there a difference in average number children picked up late before there is a fine, during the time the fine is implemented, and after there the fine is removed? \[H_0: \mu_{Before} = \mu_{During} = \mu_{After} = \mu\] \[H_a: \mu_i \neq \mu \ \text{for at least one}\ i\in\{1 = Before,\ 2 = During,\ 3 = After\}\]

Will enforcing a fine and then taking it away have a different effect on the number of children picked up late between the control group and the fined group?

\[H_0: \text{The difference in the mean number of children picked up late} \\ \ \ \ \ \ \ \ \ \text{between the different groups is the same for each fine period.} \]

\[H_a: \text{The difference in the mean number of children picked up late} \\ \ \ \ \ \text{between the different groups differs between fine periods.} \]

The significance level for each of these hypothesis tests will be set at:

\[\alpha = 0.05\]

Study Details

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.

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”.

Late <- reshape(late,
                varying = paste("Week",1:20, sep = ""), 
                v.names = "No.ofLateChildren",
                timevar = "Week", 
                times = 1:20, 
                idvar = "Center",
                new.row.names = 1:200,
                direction = "long")
pander(Late)
Treatment Center No.ofChidren Week No.ofLateChildren
Fine 1 37 1 8
Fine 2 35 1 6
Fine 3 35 1 8
Fine 4 34 1 10
Fine 5 33 1 13
Fine 6 28 1 5
Control 7 35 1 7
Control 8 34 1 12
Control 9 34 1 3
Control 10 32 1 15
Fine 1 37 2 8
Fine 2 35 2 7
Fine 3 35 2 9
Fine 4 34 2 3
Fine 5 33 2 12
Fine 6 28 2 8
Control 7 35 2 10
Control 8 34 2 9
Control 9 34 2 4
Control 10 32 2 13
Fine 1 37 3 7
Fine 2 35 3 3
Fine 3 35 3 8
Fine 4 34 3 14
Fine 5 33 3 9
Fine 6 28 3 7
Control 7 35 3 12
Control 8 34 3 14
Control 9 34 3 9
Control 10 32 3 13
Fine 1 37 4 6
Fine 2 35 4 5
Fine 3 35 4 9
Fine 4 34 4 9
Fine 5 33 4 13
Fine 6 28 4 5
Control 7 35 4 6
Control 8 34 4 18
Control 9 34 4 3
Control 10 32 4 12
Fine 1 37 5 8
Fine 2 35 5 2
Fine 3 35 5 3
Fine 4 34 5 6
Fine 5 33 5 15
Fine 6 28 5 5
Control 7 35 5 4
Control 8 34 5 10
Control 9 34 5 3
Control 10 32 5 10
Fine 1 37 6 9
Fine 2 35 6 11
Fine 3 35 6 5
Fine 4 34 6 24
Fine 5 33 6 10
Fine 6 28 6 9
Control 7 35 6 13
Control 8 34 6 11
Control 9 34 6 5
Control 10 32 6 9
Fine 1 37 7 9
Fine 2 35 7 14
Fine 3 35 7 15
Fine 4 34 7 8
Fine 5 33 7 27
Fine 6 28 7 12
Control 7 35 7 7
Control 8 34 7 6
Control 9 34 7 9
Control 10 32 7 15
Fine 1 37 8 12
Fine 2 35 8 9
Fine 3 35 8 18
Fine 4 34 8 22
Fine 5 33 8 28
Fine 6 28 8 14
Control 7 35 8 8
Control 8 34 8 15
Control 9 34 8 5
Control 10 32 8 15
Fine 1 37 9 13
Fine 2 35 9 16
Fine 3 35 9 16
Fine 4 34 9 22
Fine 5 33 9 35
Fine 6 28 9 19
Control 7 35 9 5
Control 8 34 9 14
Control 9 34 9 2
Control 10 32 9 15
Fine 1 37 10 13
Fine 2 35 10 12
Fine 3 35 10 14
Fine 4 34 10 19
Fine 5 33 10 10
Fine 6 28 10 17
Control 7 35 10 12
Control 8 34 10 13
Control 9 34 10 7
Control 10 32 10 10
Fine 1 37 11 15
Fine 2 35 11 10
Fine 3 35 11 20
Fine 4 34 11 25
Fine 5 33 11 24
Fine 6 28 11 14
Control 7 35 11 3
Control 8 34 11 7
Control 9 34 11 6
Control 10 32 11 17
Fine 1 37 12 13
Fine 2 35 12 14
Fine 3 35 12 18
Fine 4 34 12 18
Fine 5 33 12 32
Fine 6 28 12 13
Control 7 35 12 5
Control 8 34 12 12
Control 9 34 12 6
Control 10 32 12 12
Fine 1 37 13 14
Fine 2 35 13 14
Fine 3 35 13 25
Fine 4 34 13 23
Fine 5 33 13 29
Fine 6 28 13 10
Control 7 35 13 6
Control 8 34 13 9
Control 9 34 13 9
Control 10 32 13 13
Fine 1 37 14 16
Fine 2 35 14 16
Fine 3 35 14 22
Fine 4 34 14 22
Fine 5 33 14 29
Fine 6 28 14 15
Control 7 35 14 13
Control 8 34 14 9
Control 9 34 14 4
Control 10 32 14 11
Fine 1 37 15 14
Fine 2 35 15 12
Fine 3 35 15 27
Fine 4 34 15 24
Fine 5 33 15 26
Fine 6 28 15 14
Control 7 35 15 7
Control 8 34 15 17
Control 9 34 15 9
Control 10 32 15 14
Fine 1 37 16 15
Fine 2 35 16 17
Fine 3 35 16 19
Fine 4 34 16 17
Fine 5 33 16 31
Fine 6 28 16 16
Control 7 35 16 4
Control 8 34 16 8
Control 9 34 16 2
Control 10 32 16 17
Fine 1 37 17 16
Fine 2 35 17 14
Fine 3 35 17 20
Fine 4 34 17 15
Fine 5 33 17 26
Fine 6 28 17 6
Control 7 35 17 7
Control 8 34 17 5
Control 9 34 17 3
Control 10 32 17 12
Fine 1 37 18 13
Fine 2 35 18 10
Fine 3 35 18 23
Fine 4 34 18 23
Fine 5 33 18 35
Fine 6 28 18 12
Control 7 35 18 10
Control 8 34 18 11
Control 9 34 18 8
Control 10 32 18 9
Fine 1 37 19 15
Fine 2 35 19 14
Fine 3 35 19 23
Fine 4 34 19 25
Fine 5 33 19 29
Fine 6 28 19 17
Control 7 35 19 4
Control 8 34 19 8
Control 9 34 19 3
Control 10 32 19 15
Fine 1 37 20 17
Fine 2 35 20 15
Fine 3 35 20 22
Fine 4 34 20 18
Fine 5 33 20 28
Fine 6 28 20 13
Control 7 35 20 6
Control 8 34 20 13
Control 9 34 20 5
Control 10 32 20 13


Analysis

The graph below shows a difference in the number of children with late parents between the fined group and the control group. Interestingly the fine seems to have the opposite of the desired effect and has made the day-care centers’ problems more frequent.

Late$Period <- cut(Late$Week,
                 c(0,4,16,20),
                 labels = c("Before Fine","During Fine","After Fine"))
          
Late.aov <- aov(No.ofLateChildren ~ 
                  (Treatment + Period + Treatment:Period),
                data = Late)

xyplot(No.ofLateChildren ~ Treatment, data = Late, type = c("p","a"),
       jitter.x = TRUE, cex = .6, 
       pch = 16,
       col = "forestgreen",
       main = "Late Fine Increases Lateness Frequency",
       ylab = "Number of Children with Late Parents",
       xlab = "Treatment Group")

Late %>% 
  group_by(Treatment) %>% 
  summarise("Mean" = mean(No.ofLateChildren),
            SD = sd(No.ofLateChildren),
            n = n()) %>% 
  pander(caption = "Numerical Summary of Above")
Numerical Summary of Above
Treatment Mean SD n
Control 9.188 4.189 80
Fine 15.21 7.397 120

The graph below shows an increase in the number of children with late parents once a fine was enforced. After the fine was removed, the mean lateness frequency did not lower back to before but increased slightly.

xyplot(No.ofLateChildren ~ Period, data = Late, type = c("p","a"),
       jitter.x = TRUE, cex = .6, pch = 16,
       col = "firebrick2",
       main = "Increase in Lateness Frequency Throughout Fine Periods ",
       ylab = "Number of Children with Late Parents",
       xlab = "Fine Period")

Late %>% 
  group_by(Period) %>% 
  summarise("Mean" = mean(No.ofLateChildren),
            SD = sd(No.ofLateChildren),
            n = n()) %>% 
  pander(caption = "Numerical Summary of Above")
Numerical Summary of Above
Period Mean SD n
Before Fine 8.8 3.695 40
During Fine 13.56 7.068 120
After Fine 14.53 7.733 40

The graph below further shows the relationship of the control group and the fine group along with the changes in the frequency of lateness while the fine was implemented and then removed. As with the previous graph, parents show no signs of returning to their previous habits after the fine was lifted.

xyplot(No.ofLateChildren ~ Period, data = Late, type = c("p", "a"),
       groups = Treatment,
       par.settings = list(superpose.symbol = list(pch = 16, cex = .6)),
       auto.key = list(corner = c(.02,.96)),
       jitter.x = TRUE, 
       main = "Fine Group Lateness Frequency Greater Than
Control Group Lateness Frequency Throughout Fine Periods",
       ylab = "Number of Children with Late Parents",
       xlab = "Fine Period")

Late %>% 
  group_by(Treatment, Period) %>% 
  summarise("Mean" = mean(No.ofLateChildren),
            SD = sd(No.ofLateChildren),
            n = n()) %>% 
  pander(caption = "Numerical Summary of Above")
Numerical Summary of Above
Treatment Period Mean SD n
Control Before Fine 10 4.442 16
Control During Fine 9.229 4.249 48
Control After Fine 8.25 3.804 16
Fine Before Fine 8 2.934 24
Fine During Fine 16.44 7.118 72
Fine After Fine 18.71 6.805 24

To see if the differences observed in the previous graphs are significant, we will perform an analysis of variance. To know the valididty of our results, we must check the requirements of the ANOVA test. See the graphs below. As supported by the Q-Q plot on the right, we will assume normality of the error terms. However, there does seem to be an increase in the spread of the residuals in the graphic on the left which makes constant variance questionable. With this in mind, we will perform the ANOVA

par(mfrow = c(1,2))
plot(Late.aov, which = 1:2, pch = 16, cex = .6, col = "#001878")

#mtext("Requirements for ANOVA Satisfied", cex = 1.2,outer = TRUE, line = -2)
pander(Late.aov)
Analysis of Variance Model
  Df Sum Sq Mean Sq F value Pr(>F)
Treatment 1 1740 1740 54.25 4.925e-12
Period 2 828 414 12.91 5.456e-06
Treatment:Period 2 847.7 423.9 13.22 4.162e-06
Residuals 194 6222 32.07 NA NA

The table above shows gives the results of the ANOVA and our p-values for each of our hypothesis tests. We conclude that the difference between the control group and the fine group is significant \((p = 4.925 \times 10^{-12})\), that the differences in the 3 fine periods are significant \((p = 5.456 \times 10^{-6})\), and that the difference in the mean number of children picked up late between the different groups differs significantly between fine periods. \((p = 4.162 \times 10^{-6})\). This means for each hypothesis test we have sufficient evidence to reject the null hypothesis.


Conclusion

The results of this study have shown that a late fine for parents picking up their children is not only ineffective but also counter-productive for private day-care centers in Haifa, Israel. Although we do not know the exact reason why a fine would cause these parents to be late even more than usual and worsen the situation, it is very likely that the amount of the fine was not large enough to entice parents to be more punctual. More research exploring using a different fine amount in various groups would likely have promising results.