Giving in the Church of England
These data record the amount of annual giving per church member in a sample of 20 dioceses in the Church of England. Three other potentially relevant factors, employment rate, the percentage of the population on the electoral roll of the church and the percentage of the population who usually attend church, are also recorded.
Pickering, J. (1983). An analysis of giving in the Church of England. Applied Economics 17, 619-32.
The aim of this study is to identify the predictors which are associated with giving to the church.
cofe<-read.csv("week8/Lecture15/cofe.CSV")
library(GGally)
ggpairs(cofe) +
theme(plot.background = element_rect(
fill = "transparent",
colour = NA,
size = 1))
The plots show the response of annual giving against each explanatory variable. There is a weak but positive relationship between the employment rate of the diocese and the level of giving. This is not surprising since presumably people from prosperous regions are generally able to give more. The negative relationship between giving and the proportion of people on the electoral roll, and between giving and the proportion of people who usually attend on Sundays, is surprising. Possible explanations are that (i) where more people attend, there is less perceived need to give any money, and (ii) large congregations may tend to consist of a committed core plus a less committed fringe, who correspondingly give little.
The scatterplot matrix also shows a strong relationship between electoral roll and attendance. These two variables are measuring the same thing, namely “attachment” to the church. We shall therefore leave one of them (electoral roll) out of the model.
Regression Analysis: Annual giving versus Employment rate and % of Attendance
##
## Call:
## lm(formula = giving ~ employ + attend, data = cofe)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.749 -5.011 -1.034 3.411 17.162
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -88.7299 54.3958 -1.631 0.12124
## employ 1.8375 0.6552 2.804 0.01219 *
## attend -9.1965 2.4408 -3.768 0.00153 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.1 on 17 degrees of freedom
## Multiple R-squared: 0.4762, Adjusted R-squared: 0.4146
## F-statistic: 7.729 on 2 and 17 DF, p-value: 0.004098## Analysis of Variance Table
##
## Response: giving
## Df Sum Sq Mean Sq F value Pr(>F)
## employ 1 63.58 63.58 1.2615 0.276985
## attend 1 715.55 715.55 14.1963 0.001534 **
## Residuals 17 856.87 50.40
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Regression Analysis: Annual giving versus Employment
##
## Call:
## lm(formula = giving ~ employ, data = cofe)
##
## Residuals:
## Min 1Q Median 3Q Max
## -11.206 -6.389 -1.495 6.209 22.570
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.2907 66.0132 -0.141 0.890
## employ 0.6442 0.7551 0.853 0.405
##
## Residual standard error: 9.346 on 18 degrees of freedom
## Multiple R-squared: 0.03886, Adjusted R-squared: -0.01453
## F-statistic: 0.7279 on 1 and 18 DF, p-value: 0.4048Regression Analysis: Annual giving versus % of Attendance
##
## Call:
## lm(formula = giving ~ attend, data = cofe)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9.822 -5.472 -2.612 5.631 22.234
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 62.897 7.033 8.944 4.84e-08 ***
## attend -5.888 2.511 -2.344 0.0307 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 8.344 on 18 degrees of freedom
## Multiple R-squared: 0.2339, Adjusted R-squared: 0.1914
## F-statistic: 5.496 on 1 and 18 DF, p-value: 0.03073Confidence intervals
Confidence intervals can be produced from each analyses in R for the
parameters in each of the models above, using the confint command.
Annual giving versus Employment rate and % of Attendance
## 2.5 % 97.5 %
## (Intercept) -203.494988 26.035188
## employ 0.455149 3.219845
## attend -14.346174 -4.046839ggcoef(giving.lm,vline_color = "red",
vline_linetype = "solid",
errorbar_color = "blue",
errorbar_height = .25,exclude_intercept = TRUE)+
theme(plot.background = element_rect(
fill = "transparent",
colour = NA,
size = 1))
ggcoef(giving.lm,vline_color = "red",
vline_linetype = "solid",
errorbar_color = "blue",
errorbar_height = .25,exclude_intercept = TRUE) +
theme(plot.background = element_rect(
fill = "transparent",
colour = NA,
size = 1))
Annual giving versus Employment rate
## 2.5 % 97.5 %
## (Intercept) -147.9793277 129.3980
## employ -0.9421889 2.2306## 2.5 % 97.5 %
## (Intercept) 48.12165 77.6718318
## attend -11.16383 -0.6115353These intervals can also be produced by hand using the standard error information.
In order to examine whether employment rate and % of attendance are related to annual giving, we can construct C.I.s for their regression coefficients from the full model where we have fitted both explanatory variables:
Employment rate:
\[\begin{aligned} 1.837 &\pm t(17;0.975) \times 0.655\\ 1.837 &\pm 2.110 \times 0.655\\ 1.837 &\pm 1.38\\ (0.46 &, 3.22)\end{aligned}\]
% of Attendance:
\[\begin{aligned} -9.197 &\pm 2.11 \times 2.441\\ -9.197 &\pm 5.151\\ (-14.348 &, -4.046)\end{aligned}\]
Neither of these C.I.s contain zero, so we have clear evidence that employment rate and % of attendance are both related to annual giving in addition to one another.
This picture emerged from fitting the full model, with both employment rate and % of attendance involved. If, instead, we had fitted the simple model
\[E(Y) = \alpha+\beta (\text{employment rate})\]
the R output shows that a 95% C.I. for \(\beta\) is given by
\[\begin{aligned} 0.644 &\pm t(18;0.975) \times 0.755\\ 0.644 &\pm 2.101 \times 0.755\\ 0.644 &\pm 1.586\\ (-0.942,&2.230)\end{aligned}\]
Since this CI includes 0, this gives no indication of a relationship between employment rate and annual giving.
For the model with the % of attendance as the only variable, we have
\[E(Y) = \alpha+\gamma (\text{\% of attendance}),\]
and a 95% C.I. for \(\gamma\) is:
\[\begin{aligned} -5.888 &\pm t(18;0.975) \times 2.511\\ -5.888 &\pm 2.101 \times 2.511\\ -5.888 &\pm 5.276\\ (-11.164&, -0.612)\end{aligned}\]
which does give clear indication of a relationship.
It is strange that the model
\[E(Y) = \alpha+\beta (\text{employment rate})\]
produces no evidence that \(\beta \neq 0\), whereas the model
\[E(Y) = \alpha+\beta (\text{employment rate})+\gamma (\text{\% of attendance})\]
does clearly suggest \(\beta \neq 0\).
What is happening is that employment rate and % of attendance are working in opposite directions (employment, positive; attend, negative) and that when we include only employment rate in the model, the effect of attendance % obscures the relationship between annual giving and employment rate. This example gives a clear illustration of the advantages of starting analyses with the full model.