The best way to create a predictive model is to begin with a complete data set. Split the data into two seperate datasets: the training set and the testing set, with the training set greater than or equal to 50% of the data. We will use the training set to build a model best predictor of the data. Lastly, we will test the built model on the testing data set, and assess how good the model is in predicting the actual values.

The concept may seem overwhelming, but we already have almost all of the tools to perform this analysis.

### Setup

Remember to always set your working directory to the source file location. Go to ‘Session’, scroll down to ‘Set Working Directory’, and click ‘To Source File Location’. Read carefully the below and follow the instructions to complete the tasks and answer any questions. Submit your work to RPubs as detailed in previous notes.

### Note

For your assignment you may be using different data sets than what is included here. Always read carefully the instructions on Sakai. Tasks/questions to be completed/answered are highlighted in larger bolded fonts and numbered according to their particular placement in the task section.

##### PART A: TRAINING SET & MODEL BUILDING

The first half of this lab focus on building a model using the training data. The data we will be using was obtained from Kaggle site[1] and is in reference to world university rankings [2] . The data looks at university world scoring based on different ranking criteria such as rank for quality of education, rank for quality of faculty, and rank for patents. We begin by reading the training data set ‘universityrank_training.csv’ file, and checking the header lines to make sure the data is read correctly.

traindata = read.csv(file="data/universityrank_training.csv", header=TRUE)
head(traindata)
##   world_rank                           institution        country
## 1          1                    Harvard University            USA
## 2          2 Massachusetts Institute of Technology            USA
## 3          3                   Stanford University            USA
## 4          4               University of Cambridge United Kingdom
## 5          5    California Institute of Technology            USA
## 6          6                  Princeton University            USA
##   national_rank quality_of_education alumni_employment quality_of_faculty
## 1             1                    7                 9                  1
## 2             2                    9                17                  3
## 3             3                   17                11                  5
## 4             1                   10                24                  4
## 5             4                    2                29                  7
## 6             5                    8                14                  2
##   publications influence citations broad_impact patents  score year
## 1            1         1         1           NA       5 100.00 2012
## 2           12         4         4           NA       1  91.67 2012
## 3            4         2         2           NA      15  89.50 2012
## 4           16        16        11           NA      50  86.17 2012
## 5           37        22        22           NA      18  85.21 2012
## 6           53        33        26           NA     101  82.50 2012

Next, we extract the two columns of interest and call them properly so we can easily refer to them later in the code.

patent_train = traindata$patents score_train = traindata$score

The first model we will build is a simple linear model. We will use the patents ranking variable to predict the university score. To better understand the data, the lower the patents ranking number the better it is. A value of 1 is a top rank for patents and represent the highest category in terms of number of patents owned by the particular academic institution. On the other hand the higher the calculated total score the better, as reflected by the world rank number. A value of 100 is a perfect score.

linear_train = lm(score_train ~ patent_train)
summary(linear_train)
##
## Call:
## lm(formula = score_train ~ patent_train)
##
## Residuals:
##    Min     1Q Median     3Q    Max
## -9.876 -4.014 -1.121  1.515 45.470
##
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)
## (Intercept)  54.6406641  0.3859848  141.56   <2e-16 ***
## patent_train -0.0157607  0.0008291  -19.01   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 7.539 on 1197 degrees of freedom
## Multiple R-squared:  0.2319, Adjusted R-squared:  0.2313
## F-statistic: 361.4 on 1 and 1197 DF,  p-value: < 2.2e-16
plot(patent_train,score_train)
abline(linear_train, col="blue", lwd=2)

##### Consider now building a non-linear quadratic model. Follow the steps in lab07 to derive a quadratic model similar to what we did for costs versus servers. #### TASK 1: First, define a new variable which is the squared value of patent_train (defined above)

patent_train2 = patent_train^2

quad_train = lm(score_train ~ patent_train + patent_train2)

#### TASK 3: Third, publish the summary statistics.

summary (quad_train)
##
## Call:
## lm(formula = score_train ~ patent_train + patent_train2)
##
## Residuals:
##     Min      1Q  Median      3Q     Max
## -13.556  -2.347  -0.578   1.301  40.842
##
## Coefficients:
##                 Estimate Std. Error t value Pr(>|t|)
## (Intercept)    5.966e+01  4.973e-01  119.97   <2e-16 ***
## patent_train  -6.250e-02  3.321e-03  -18.82   <2e-16 ***
## patent_train2  5.975e-05  4.130e-06   14.47   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6.957 on 1196 degrees of freedom
## Multiple R-squared:  0.3463, Adjusted R-squared:  0.3452
## F-statistic: 316.8 on 2 and 1196 DF,  p-value: < 2.2e-16

#### TASK 4: Based on the values of R-Squared and Adj R-Squared for both the linear and the quadratic select the best predictive model. Explain your argument.

The quadratic r-squared is .3463 and its adjusted r-squared is .3452, and the linear r-squared is .2319 and adjusted r-squared is .2313.
Therefore the best predictive model is the quadratic. In multiple regression analysis the "Adjusted R squared" gives an idea of how the model generalises.
In an ideal situation, it is preferable that its value is as close as possible to the value of "R squared"

##### PART B: TESTING SET & MODEL EVALUATION

The second half of predictive modeling is about testing the model using a different data set called the testing data. Again we must first read the testing data set, and make sure the dataset is read propertly.

testdata = read.csv("data/universityrank_testing.csv", header=TRUE)
head(testdata)
##   world_rank                           institution        country
## 1       1000                    Yanbian University          China
## 2          1                    Harvard University            USA
## 3          2                   Stanford University            USA
## 4          3 Massachusetts Institute of Technology            USA
## 5          4               University of Cambridge United Kingdom
## 6          5                  University of Oxford United Kingdom
##   national_rank quality_of_education alumni_employment quality_of_faculty
## 1            84                  355               478                210
## 2             1                    1                 1                  1
## 3             2                    9                 2                  4
## 4             3                    3                11                  2
## 5             1                    2                10                  5
## 6             2                    7                13                 10
##   publications influence citations broad_impact patents  score year
## 1          890       790       800         1000     737  44.18 2014
## 2            1         1         1            1       3 100.00 2015
## 3            5         3         3            4      10  98.66 2015
## 4           15         2         2            2       1  97.54 2015
## 5           11         6        12           13      48  96.81 2015
## 6            7        12         7            9      15  96.46 2015

We extract again the two columns of interest, in reference this time to the testing data set, and call them accordingly.

patent_test = testdata$patents score_test = testdata$score

We are ready now to check if the derived models are actually good predictive models. First we calculate the predicted test data score using the linear models derived earlier.

# Calculate the predicted test data score
score_predict1 = coef(linear_train)[1] + coef(linear_train)[2]*patent_test

For a visual representation we can plot the actual testing data, and overlay the predicted values

# Plot the actual values for patent and score as observed in the testing data set
plot(patent_test, score_test, main='Test Data -- Score vs Patent')
# Overlay the predicted values as calculated from the linear model and derived using the training model
par(new=TRUE, xaxt="n", yaxt="n", ann=FALSE)
# The red color is used to distinguish the predicted values which, because of the linear model, will fit exactly a line
plot(patent_test, score_predict1, col="red")

A better way to qualify the goodness of a predictive model is to scatter plot the actual values against the predicted values. In a perfect predictive model the points will line up along the diagonal line. This is rarely the case, if ever!

#Plot predicted values from the linear model versus actual values form the test data
plot(score_test, score_predict1, xlab='Actual', ylab='Predict', main='Linear Model -- Predict vs Actual Test')

From the plot we can easily see that most of the predicted values versus actual are far from the diagoonal line. In many cases this is fine. Finally, to quantify the goodness of a model, we need to calculate the Root Mean Square Error (RMSE).

#Calculate RMSE for Linear Model
error1 = sum((score_predict1 - score_test)^2)/length(score_test)
rmse1 = sqrt(error1)
rmse1
## [1] 5.956123

It is hard to judge the goodness of the number unless we compare to other possibilities. Of course a perfect scenario will have zero RMSE. We now need to repeat the above calculations for the non-linear quadratic model.

#### TASK 5: Calculate score_predict2 based on the quadratic model and the patent test data. You need to refer again to the coefficients of the quadratic model derived earlier and the actual patent values obtained from the testing data.

score_predict2 = coef(quad_train)[1]+coef(quad_train)[2]*patent_test+coef(quad_train)[3]*patent_test^2
summary(score_predict2)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max.
##   43.31   44.25   47.02   47.91   50.55   59.60

For a visual representation, similar to the linear model, we need to do the following.

#### TASK 6: Plot the score versus patent actual test data and overlay the predictied values as calculated in TASK 5.

# Plot the actual values for patent and score as observed in the testing data set. Write the code below:

plot(patent_test,score_test,main='Test Data--Score vs Patent')

# Overlay the predicted values as calculated from the quadratic model and derived using the training model. Write the code below:
par(new=TRUE,xaxt="n",yaxt="n", ann=FALSE)

# The green color is used to distinguish the predicted values which, because of the quadratic model, will in this case fit exactly a parabola. Write the code below:
plot(patent_test, score_predict2, col="green")

#### TASK 7: Plot the predicted values against the true values. Label properly.

#Plot the predicted values form the quadratic model versus the actual values from the test data. Write the code below:
plot(score_test, score_predict2, xlab='Actual', ylab = 'Predict', main = 'Quadratic Model--Predict vs Actual Test')

#### TASK 8: By looking at the scatterplots for the linear and quadratic models are you able to tell which model is better? Elaborate.

The quadratic model because it seems to follow the line between actual and predicted more closely. A better way to quantify the goodness of the model is to calculate again the RMSE.

#### TASK 9: Calculate the root mean square error (RMSE) for the quadratic model.

#Calculate RMSE  for Quadratic Model. Write the code below:
error2=sum((score_predict2 -score_test)^2)/length(score_test)
rmse2=sqrt(error2)
rmse2
## [1] 5.684821