Kreis Final

## Warning: package 'gridExtra' was built under R version 3.2.5

Part 1 - Introduction:

Research Question:

Is the percentage of GDP per capital spent on education strongly correlated to Educational outcomes based on educational results among participating countries countries? This question, which can of course not be fully answered here, will help to determine whether a country’s relative investment in education is correlated to improved educational outcomes or if there may be other variables at play. We will also look closer to see if the results are similar for, the typically more wealthy, OECD member countries.

Null hypothesis: There is no statistically significant correlation between scores on the PISA Educational Assessment and the percentage of GDP per capital spent on education.

Alternative Hypothesis: There is statistical evidence that the GDP spent per capital has influence on PISA scores.

Part 2 - Data:

What are the cases and how many are there/ Data Source:

The World Bank has data for approximately 3,000 variables for 214 economies and data has been collected annually since 1970. I am specifically comparing mean results in PISA assessments for 2012. Test results are compared to separate data from the World Bank on government expenditure per primary student as a percentage of GDP per capital in the same year, if possible, or 2011 if 2012 measures were not taken.
After removing observations where either the PISA test results or GDP values (in 2011 and 2012) were not recorded, there are 139 observations (45 in math 47 in science and 47 in reading)

Describe the method of data collection:

According to the website, they use official country specific sources

What type of study is this (observational/experiment)?

This is an observational study

What is the Response Variable?

The response variable is the Mean performance on the reading, math and science scale (PISA test results). This is a numerical variable.

What is the Explanatory Variable?

The explanatory variable is the percentage of GDP spent on education for each country being tested. This is a numerical variable.

Scope of Inference

This analysis is based on an observational study and, as such, this research will not lead to any causal claims. There are numerous variables at play when evaluating educational outcomes, including levels of poverty, whether the country is at war(particularly within its own borders), parental involvement, cultural influences, social/economic class stratification, nutrition & health among countless others. In the future I may investigate changes in PISA test results from 2000 to 2012 and changes in expenditures over that same period.
Initially, I plotted data for all of the Pisa test scores for each country for the Pisa testing years from 2000 to 2012 (test results for every three years) but I became worried about whether using data for the same country over 5 time periods would lead to dependence. I elected instead to go with the most recent period 2012.

Initial Data Manipulation

After initially looking at all Pisa assessment results for which we have a corresponding GDP spending figure we can now look more closely at the OECD member countries to see if there is a large difference.

Part 3 - Exploratory data analysis:

All Countries

The minimum percentage of GDP per capital per student spent on education was 9.22% (Peru) compared to a maximum of 70.65% (Lithuania). The next closest to Lithuania is Serbia at 50.95%, both of these countries are significantly above the Median GDP expenditure of 20.56%

x <- GDP %>% arrange(desc(gdp))
head(x)

##   Country.Code  year      gdp
## 1          LTU X2012 70.64836
## 2          SRB X2012 50.95265
## 3          MDA X2012 39.31168
## 4          CYP X2012 32.57038
## 5          UKR X2012 32.38915
## 6          TLS X2012 31.89363

rm(x)

Percentage of GDP appears appears to be mostly normal with some significant outliers on the upper x axis. Test scores in each category appear to be nearly normally distributed with a right skew. The limited number of data points makes it difficult to discern. Please see the summary statistics for science math and reading scores below:

#summary of GDP measures
summary(scienceA$gdp)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   9.221  17.120  20.560  21.780  23.810  70.650

#summary of the various scores
summary(scienceA$score)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   373.1   445.7   495.7   482.2   515.5   554.9

summary(mathA$score)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   368.1   448.9   489.8   476.0   505.5   561.2

summary(readingA$score)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   384.2   443.9   488.5   476.7   508.4   544.6

a1<- ggplot(scienceA, aes(score)) + geom_histogram(binwidth=10, color='dodgerblue1') + ggtitle('Science All Countries')
a2<- ggplot(mathA, aes(score)) + geom_histogram(binwidth=10, color='dodgerblue1') + ggtitle('Math All Countries')
a3<- ggplot(readingA, aes(score)) + geom_histogram(binwidth=10, color='dodgerblue1') + ggtitle('Reading All Countries')
a4 <- ggplot(scienceA, aes(gdp)) + geom_histogram(binwidth=2, color='green') + ggtitle('GDP All Countries')
grid.arrange(a1, a2, a3, a4)

OECD Member Countries Only

The percentage of GDP per student per capital spent in the OECD member countries in 2012 ranged from 14.60% (Mexico) to 29.95(Slovenia) with a median value of 20.71. The distribution of GDP appears to be normal
The distribution of all scores also appears to be normal with a right skew, please see the summary statistics below:

#summary of GDP measures
summary(science$gdp)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   14.60   18.84   20.71   21.32   23.82   29.95

#summary of the various scores
summary(science$score)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   414.9   491.8   501.9   501.5   520.0   546.7

summary(math$score)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   413.3   484.6   497.0   494.2   511.5   536.4

summary(reading$score)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   423.6   487.8   496.9   495.6   510.7   538.1

p1<- ggplot(science, aes(score)) + geom_histogram(binwidth=5, color='dodgerblue1') + ggtitle('science')
p2<- ggplot(math, aes(score)) + geom_histogram(binwidth=5, color='dodgerblue1') + ggtitle('math')
p3<- ggplot(reading, aes(score)) + geom_histogram(binwidth=5, color='dodgerblue1') + ggtitle('reading')
p4 <- ggplot(science, aes(gdp)) + geom_histogram(binwidth=1, color='green') + ggtitle('gdp')
grid.arrange(p1, p2, p3, p4)

Part 4 - Inference:

There does appear to be a positive relationship between spending and educational attainment; however, it does not appear to be particularly strong.

*The residuals are nearly normal

*I don’t believe that the data meets the condition for roughly constant variability around the least squares line.

# This plot is for a visual only because each test score has three scores associated with it.  To do linear regression we need to break out each score.
ggplot(combined) + aes(x = gdp, y = score, color = Indicator.Code) + geom_point()

All Countries in Sample

g1 <- ggplot(mathA) + aes(x = gdp, y = score) + geom_point(color = 'dodgerblue2') + ggtitle('math') + geom_smooth(method = "lm", se = FALSE)


g2 <- ggplot(scienceA) + aes(x = gdp, y = score) + geom_point(color = 'red') + ggtitle('science')+ geom_smooth(method = "lm", se = FALSE)


g3 <- ggplot(readingA) + aes(x = gdp, y = score) + geom_point(color = 'green') + ggtitle('reading') + geom_smooth(method = "lm", se = FALSE)
grid.arrange(g1, g2, g3)

MATH 3.5 % of the variability in math scores for all of the countries in our sample can be explained by the percentage of GDP per capital spent on education. The p-value of 0.2187 indicates that there is only 21.87% chance we arrived at this conclusion by chance. In this case we will not reject the null hypothesis that there is no statistically significant correlation between scores on the PISA Educational Assessment and the percentage of GDP per capital spent on education.

a1 <- lm(score ~ gdp, data = mathA)
summary(a1)

## 
## Call:
## lm(formula = score ~ gdp, data = mathA)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -96.31 -42.39  13.16  31.36  92.23 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 455.8906    17.6851  25.778   <2e-16 ***
## gdp           0.9246     0.7407   1.248    0.219    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 48.55 on 43 degrees of freedom
## Multiple R-squared:  0.03497,    Adjusted R-squared:  0.01253 
## F-statistic: 1.558 on 1 and 43 DF,  p-value: 0.2187

Science 3.2 % of the variability in science scores for all of the countries in our sample can be explained by the percentage of GDP per capital spent on education. The p-value of 0.2187 indicates that there is only a 22.78% chance our findings are due to chance. In this case we will not reject the null hypothesis that there is no statistically significant correlation between scores on the PISA Educational Assessment and the percentage of GDP per capital spent on education.

a2 <- lm(score ~ gdp, data = scienceA)
summary(a2)

## 
## Call:
## lm(formula = score ~ gdp, data = scienceA)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -98.51 -33.52  14.49  31.41  79.11 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 463.8244    16.5292  28.061   <2e-16 ***
## gdp           0.8459     0.6918   1.223    0.228    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 46.59 on 45 degrees of freedom
## Multiple R-squared:  0.03215,    Adjusted R-squared:  0.01065 
## F-statistic: 1.495 on 1 and 45 DF,  p-value: 0.2278

Reading 2.64 % of the variability in science scores for all of the countries in our sample can be explained by the percentage of GDP per capital spent on education. The p-value of 0.275 indicates that there is only a 27.5% chance our findings are due to chance. In this case we will not reject the null hypothesis that there is no statistically significant correlation between scores on the PISA Educational Assessment and the percentage of GDP per capital spent on education.

a3 <- lm(score ~ gdp, data = readingA)
summary(a3)

## 
## Call:
## lm(formula = score ~ gdp, data = readingA)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -83.76 -34.31  13.34  31.09  73.22 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 461.4919    15.0602  30.643   <2e-16 ***
## gdp           0.6965     0.6303   1.105    0.275    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 42.45 on 45 degrees of freedom
## Multiple R-squared:  0.02642,    Adjusted R-squared:  0.004783 
## F-statistic: 1.221 on 1 and 45 DF,  p-value: 0.275

Model Diagnostics

The extreme outliers are noteworthy and should possibly be removed. The tests for these countries indicate that fitting a least squares line to this data, at least as it is, may be unwise. The variability around the line in our qqnorm plot is relatively consistent for science and reading but has a large deviation from the line on math.

r1 <- ggplot(mathA) + aes(x = gdp, y = a1$residuals) + geom_point(color = 'dodgerblue2') + ggtitle('math') + geom_smooth(method = "lm", se = FALSE)


r2 <- ggplot(scienceA) + aes(x = gdp, y = a2$residuals) + geom_point(color = 'red') + ggtitle('science')+ geom_smooth(method = "lm", se = FALSE)


r3 <- ggplot(readingA) + aes(x = gdp, y = a3$residuals) + geom_point(color = 'green') + ggtitle('reading') + geom_smooth(method = "lm", se = FALSE)
grid.arrange(r1, r2, r3)


s1<- ggplot(scienceA, aes(a2$residuals)) + geom_histogram(binwidth=15, color='dodgerblue1') + ggtitle('science')
s2<- ggplot(mathA, aes(a1$residuals)) + geom_histogram(binwidth=15, color='dodgerblue1') + ggtitle('math')
s3<- ggplot(readingA, aes(a3$residuals)) + geom_histogram(binwidth=15, color='dodgerblue1') + ggtitle('reading')

grid.arrange(s1, s2, s3)

qqnorm(a1$residuals) 
qqline(a1$residuals)

qqnorm(a2$residuals) 
qqline(a2$residuals)

qqnorm(a3$residuals) 
qqline(a3$residuals)

OECD Member Countries

x1 <- ggplot(math) + aes(x = gdp, y = score) + geom_point(color = 'dodgerblue2') + ggtitle('math') + geom_smooth(method = "lm", se = FALSE)


x2 <- ggplot(science) + aes(x = gdp, y = score) + geom_point(color = 'red') + ggtitle('science')+ geom_smooth(method = "lm", se = FALSE)


x3 <- ggplot(reading) + aes(x = gdp, y = score) + geom_point(color = 'green') + ggtitle('reading') + geom_smooth(method = "lm", se = FALSE)
grid.arrange(x1, x2, x3)

MATH 13.9 % of the variability in math scores for OECD countries can be explained by the percentage of GDP per capital spent on education In this case we will reject the null hypothesis that there is no statistically significant correlation between scores on the PISA Educational Assessment and the percentage of GDP per capital spent on education.The p-value of 0.0425 indicates that if the null were true there is a less than a one in twenty chance that we arrived at this conclusion by chance.

m1 <- lm(score ~ gdp, data = math)
summary(m1)

## 
## Call:
## lm(formula = score ~ gdp, data = math)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -62.312 -10.750  -0.206  17.678  40.213 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  435.241     28.091  15.494 2.89e-15 ***
## gdp            2.764      1.300   2.126   0.0425 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 25.16 on 28 degrees of freedom
## Multiple R-squared:  0.139,  Adjusted R-squared:  0.1082 
## F-statistic: 4.519 on 1 and 28 DF,  p-value: 0.04248

SCIENCE 9.09% of the variability in science scores for OECD countries can be explained by the percentage of GDP per capital spent on education. The p-value of 0.165 indicates that there is a 16.5% chance we arrived at this conclusion by chance. In this case we will not reject the null hypothesis that there is no statistically significant correlation between scores on the PISA Educational Assessment and the percentage of GDP per capital spent on education.

m2 <- lm(score ~ gdp, data = science)
summary(m2)

## 
## Call:
## lm(formula = score ~ gdp, data = science)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -70.902 -10.098  -0.001  18.229  45.530 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  451.700     30.255  14.930 7.34e-15 ***
## gdp            2.337      1.400   1.669    0.106    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 27.1 on 28 degrees of freedom
## Multiple R-squared:  0.09048,    Adjusted R-squared:  0.058 
## F-statistic: 2.785 on 1 and 28 DF,  p-value: 0.1063

READING 6.77% of the variability in reading scores for OECD countries can be explained by the percentage of GDP per capital spent on education. The p-value of 0.058 indicates that there is a 5.8% chance we arrived at this conclusion by chance. In this case we will not reject the null hypothesis that there is no statistically significant correlation between scores on the PISA Educational Assessment and the percentage of GDP per capital spent on education.

m3 <- lm(score ~ gdp, data = reading)
summary(m3)

## 
## Call:
## lm(formula = score ~ gdp, data = reading)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -60.65 -10.86   1.46  14.77  38.26 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  459.441     25.699  17.878   <2e-16 ***
## gdp            1.696      1.189   1.426    0.165    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 23.02 on 28 degrees of freedom
## Multiple R-squared:  0.06769,    Adjusted R-squared:  0.03439 
## F-statistic: 2.033 on 1 and 28 DF,  p-value: 0.165

Model Diagnostics

The OECD values exclude the main problematic outliers and as such all model diagnostics are met: -Linearity -Nearly Normal Residuals -Constant Variability -Independent Observations.

t1 <- ggplot(math) + aes(x = gdp, y = m1$residuals) + geom_point(color = 'dodgerblue2') + ggtitle('math') + geom_smooth(method = "lm", se = FALSE)


t2 <- ggplot(science) + aes(x = gdp, y = m2$residuals) + geom_point(color = 'red') + ggtitle('science')+ geom_smooth(method = "lm", se = FALSE)


t3 <- ggplot(reading) + aes(x = gdp, y = m3$residuals) + geom_point(color = 'green') + ggtitle('reading') + geom_smooth(method = "lm", se = FALSE)
grid.arrange(t1, t2, t3)



u1<- ggplot(math, aes(m1$residuals)) + geom_histogram(binwidth=15, color='dodgerblue1') + ggtitle('math')

u2<- ggplot(science, aes(m2$residuals)) + geom_histogram(binwidth=15, color='dodgerblue1') + ggtitle('science')

u3<- ggplot(reading, aes(m3$residuals)) + geom_histogram(binwidth=15, color='dodgerblue1') + ggtitle('reading')

grid.arrange(u1, u2, u3)

qqnorm(m1$residuals) 
qqline(m1$residuals)

qqnorm(m2$residuals) 
qqline(m2$residuals)

qqnorm(m3$residuals) 
qqline(m3$residuals)

Part 5 - Conclusion:

The outlier values in Lithuania and Serbia were problematic in the large data set and our conclusions may therefore be invalid. Removing those outliers would be a logical next step. According to past years of data for the world bank these extreme values do not appear to be erroneous for the two countries.
For the OECD data the correlation and statistical significance of the relationship between the expenditure per primary student as a percentage of GDP per capital does not appear to be very strong, with only the relationship to math scores being statistically significant at the 0.05 level.
One limitation of this analysis only looks at the percentage of GDP spent as a percentage of pupil. Some countries GDP is significantly higher than others, therefore the percentage does not directly indicate the amount of dollars being spent, but rather the proportional commitment of those countries to education. This warrants further more in depth investigation and a logical next step would to look at actual dollars in comparison to educational outcomes.
I have learned that this topic warrants further investigation and that even my statistically significant result has left me skeptical. In the future, I may look at other international assessments and economic measures. It may also be interesting to investigate changes in PISA test results from 2000 to 2012 and changes in expenditures over that same period to see if increases or decreases in expenditures have led to changes in performance.

References:

The World Bank GDP The World Bank PISA OECD OpenIntro