R-Code for Exam, Modules 9-13

Question 1

query_X = DATASET_NAME$CAT_VARIABLE == “X”
index_X = which(query_X)
quantities_X = DATASET_NAME$QUANT_VARIABLE[index_X]
m_quantities_X = mean(quantities_X)
sum_squares_X = sum((quantities_X - m_quantities_X)^2)

Answer 1

we have a data set with three categorical variables – X, Y and Z – and we want to find the sum of squares associated with the quantities for variable X

Question 2

query_Y = DATASET_NAME$CAT_VARIABLE == “Y”
index_Y = which(query_Y)
quantities_Y = DATASET_NAME$QUANT_VARIABLE[index_Y]
m_quantities_Y = mean(quantities_Y)
sum_squares_Y = sum((quantities_Y - m_quantities_Y)^2)
query_Z = DATASET_NAME$CAT_VARIABLE == “Z”

Answer 2

we have a data set with three categorical variables – X, Y and Z – and we want to find the sum of squares associated with the quantities for variable Y

Question 3

query_Z = DATASET_NAME$CAT_VARIABLE == “Z”
index_Z = which(query_Z)
quantities_Z = DATASET_NAME$QUANT_VARIABLE[index_Z]
m_quantities_Z = mean(quantities_Z)
sum_squares_Z = sum((quantities_Z - m_quantities_Z)^2)

Answer 3

we have a data set with three categorical variables – X, Y and Z – and we want to find the sum of squares associated with the quantities for variable Z

Question 4

within_group_ss = sum(sum_squares_X + sum_squares_Y + sum_squares_Z)

Answer 4

calculates the within group sum of squares (“within_group_ss”) for three categorical variables (X, Y and Z), whose individual sum of squares has already been calculated

Question 5

grand_mean = mean(DATASET_NAME$QUANT_VARIABLE)

Answer 5

calculates the grand mean of the quantitative variable of the dataset (saved as “grand_mean”), which represents the mean value across all the members of the categorical variable (X, Y, Z, etc.)

Question 6

among_group_ss =
(# REPS_X)(m_quantities_X - grand_mean)^2 +
(# REPS_Y)(m_quantities_Y - grand_mean)^2 +
(# REPS_Z)*(m_quantities_Z - grand_mean)^2

Answer 6

code to find the among group sum of squares for three categorical variables (X, Y and Z), given the number of replications they have in the dataset, the grand mean among the three variables, and the mean quantities of each

Question 7

total_ss = within_group_ss + among_group_ss

Answer 7

the total sum of squares equals the sum of the within group and the among group sum of squares

Question 8

table_ss =
as.table(rbind(c(“total_ss”,
“within_group_ss”,
“among_group_ss”),
c(total_ss,
within_group_ss,
among_group_ss)))

Answer 8

code which establishes a table to compare the within, among, and total sum of squares for a set of data

Question 9

anova_model =
aov(QUANTITATIVE~CATEGORICAL,
data = DATASET_NAME)

Answer 9

code to establish an ANOVA model for the data (“anova_model”) which relates a quantitative and categorical variable (“QUANTITATIVE”; “CATEGORICAL”) to a dataset (“DATASET_NAME”)

Question 10

plot(factored_variable, DATASET_NAME$QUANT_VARIABLE,
ylim = c(#, #), ylab = “BLAH BLAH”,
xlab = “BLAH BLAH”,
col = “red”/”blue”/”green” [etc.])

Answer 10

creates a box-plot of a categorical variable which has been coded as a factor variable, which allows one to see if the categories differ significantly in their expression from one another

Question 11

anova_model_1 = aov(quantitative_variable ~
factored_variable,
data = DATASET_NAME)

Answer 11

first method to establish an ANOVA model of a dataset, assuming we have already created the object “factored_variable” which relates to our variables “X”, “Y” and “Z”

Question 12

anova_model_2 = aov(quantitative_variable ~ factor(DATASET_NAME$CAT_VARIABLE, c(“X”, “Y”, “Z”),
data = DATASET_NAME)

Answer 12

second method to establish an ANOVA model of a dataset, assuming we have not yet created the object “factored_variable” for the variables “X”, “Y” and “Z”

Question 13

residuals = quantitative_variable – predict.lm(anova_model)
plot(predict.lm(anova_model), residuals, cex.lab = 1.15,
ylim = c(LOW #, HIGH #),
xlim = c(LOW #, HIGH #),
ylab = “BLAH BLAH”,
xlab = “YADA YADA”)
abline(a = 0, b = 0, lwd = 2/3, lty = “dashed”, col = “red”)

Answer 13

code we could establish to plot the residuals versus predicted values for the data set to check for the homogeneity of variance

Question 14

stdRes = rstandard(anova_model)

Answer 14

calculates the standard residuals for the ANOVA model (“anova_model”) and saves them as the object “stdRes”

Question 15

qqnorm(stdRes,
ylab=”Standardized Residuals”,
xlab=”Theoretical Quantiles”)
qqline(stdRes, col=”red”, lwd=2)

Answer 15

establishes a normal Q-Q plot for the residuals, along with the linear line of best fit

Question 16

TukeyHSD(anova_model, ordered = TRUE)

Answer 16

conducts a Tukey’s Honestly Significant Difference test on the ANOVA model in order to determine which groups in the ANOVA vary significantly from one another

Question 17

plot(TukeyHSD(anova_model, ordered = TRUE))

Answer 17

establishes a plot of the Tukey’s test of the ANOVA model – pairs of variables which don’t overlap with the dashed horizontal line at x = 0 are significantly different, while those which lie on the dashed line are significantly different

Question 18

quant_var = DATASET_NAME$QUANTITATIVE_VARIABLE
fact_var1 = factor(DATASET_NAME$CATEGORY_VARIABLE_1)
fact_var2 = factor(DATASET_NAME$CATEGORY_VARIABLE_2)
par(mfrow = c(1, 2))
plot(fact_var1, quant_var,
c(LOW #, HIGH #),
ylab = “BLAH BLAH”,
xlab = “YADA YADA”,
col = “red”)
plot(fact_var2, quant_var,
c(LOW #, HIGH #),
ylab = “BLAH BLAH”,
xlab = “YADA YADA”,
col = “blue”)

Answer 18

codes which if typed out would factor two categorical variables and establish an object for the quantitative variables, then would allow for two box-plots to be set up to compare the relationship between the first factored variable and the quantitative variable and the second factored variable and the quantitative variable

Question 19

2way_anova_model = aov(quant_var ~ fact_var1 + fact_var2)

Answer 19

code to produce a 2-way ANOVA model, given we have the quantitative variable and have factorized the two categorical variables already

Question 20

summary(2way_anova_model
[Pr(>F); fact_var1 < 0.05,
fact_var2 <0.05]
summary(1way_anova_model)
[Pr(>F); factored_variable > 0.05]

Answer 20

describes the possibility that when running a block ANOVA, the P-values for the two factor variables may be significant, but having a simpler one-factor ANOVA model which doesn’t take one of those variables into account may produce a P-value which is not significant

Question 21

library(HSAUR2)
data(“USairpollution”)

Answer 21

loads the “USairpollution” dataset from the “HSAUR2” package; “HSAUR2” is ingrained into RStudio, so you can load it without having to rely on the data sets provided by the professors

Question 22

plot(USairpollution$predays, USairpollution$SO2,
xlab = “days with precipitation”,
ylab = “sulfur emissions”)

Answer 22

creates a box-plot which relates the days with precipitation on the x-axis and the sulfur emissions on the y-axis

Question 23

pollution_model = lm(USairpollution$SO2 ~ USairpollution$predays)
abline(pollution_model,
col = “red”, lwd = 3)

Answer 23

graphs a red linear line of best fit with a width of three for the relationship between sulfur emissions and days with precipitation from the “USairpollution” data set

Question 24

predicted = predict.lm(pollution_model)
residuals = USairpollution$SO2 - predicted
plot(predicted, residuals,
xlab = “Predicted Y”,
ylab = “Residuals”)
abline(a = 0, b = 0,
col = “red”, lwd = 3, lty = “dashed”)

Answer 24

commands to establish a plot of the residuals versus the predicted values for the sulfur emissions given the number of days with precipitation (variance appears heteroscedastic).

Question 25

stdRes = rstandard(pollution_model) log_pollute_mod = lm(log(USairpollution$SO2) ~ USairpollution$predays) plot(USairpollution$predays, log(USairpollution$SO2), xlab = “days with precipitation”, ylab = “sulfur emissions (log-transformed)”) abline(log_pollute_mod, col = “red”, lwd = 3)

Answer 25

graphs a red linear line of best fit with a width of three for the relationship between the log-transformed sulfur emissions and days with precipitation from the "USairpollution" data set

Question 26

log_predicted = predict.lm(log_pollute_mod) log_residuals = log(USairpollution$SO2) - log_predicted plot(log_predicted, log_residuals, xlab = “Log-Trans Predicted Y”, ylab = “Log-Trans Residuals”) abline(a = 0, b = 0, col = “red”, lwd = 3, lty = “dashed”)

Answer 26

commands to establish a plot of the log-transformed residuals versus the predicted values for the sulfur emissions given the number of days with precipitation (variance still appears heteroscedastic).

Question 27

rainydays_sulfur = lm(USairpollution$SO2 ~ USairpollution$predays) stdRes = rstandard(rainydays_sulfur) qqnorm(stdRes, xlab = “Theoretical Quantiles”, ylab = “Standardized Residuals”) qqline(stdRes, col = “red”, lwd = 2, lty = “dashed”)

Answer 27

creates a QQ-plot for the relationship between SO2 emissions and days with precipitation

Question 28

log_stdRes = rstandard(log_pollute_mod) qqnorm(log_stdRes, xlab = “Theoretical Quantiles (Log-Transformed)”, ylab = “Standardized Residuals (Log-Transformed)”) qqline(log_stdRes, col = “red”, lwd = 2, lty = “dashed”)

Answer 28

establishes the qq-plot for the log-transformed data from the USairpollution data set

Question 29

plot(x = DATASET_NAME$VARIABLE_1, y = DATASET_NAME$VARIABLE_2) mean_var1 = mean(DATASET_NAME$VARIABLE_1) mean_var2 = mean(DATASET_NAME$VARIABLE_2) abline(v = mean_var1, col = “gray”) abline(h = mean_var2, col = “gray”)

Answer 29

commands relevant to simple regression which are used to see if two variables which are quantitative (VARIABLE_1 and VARIABLE_2) have a positive, negative or no correlation

Question 30

cor(DATASET_NAME)

Answer 30

gives the correlation between the different variables in a dataset -- each variable has a 1.000... correlation to itself but can have a corrleation of anywhere between -1.000... and 1.000... to another variable

Question 31

devi_x = DATASET_NAME$VARIABLE_1 - mean_var1 devi_y = DATASET_NAME$VARIABLE_2 - mean_var2 Pearson_numerator = sum(devi_x*devi_y)

Answer 31

manually calculates the numerator of the Pearson coefficient for VARIABLE_1 and VARIABLE_2

Question 32

sosq_devi_x = sum(devi_x^2) sosq_devi_y = sum(devi_y^2) Pearson_denominator = sqrt(sosq_devi_x*sosq_devi_y)

Answer 32

manually calculates the numerator of the Pearson coefficient for VARIABLE_1 and VARIABLE_2

Question 33

Pearson_coefficient = Pearson_numerator/Pearson_denominator

Answer 33

manually derives the Pearson coefficient, which is above zero if there is a positive correlation and below zero if there is a negative correlation

Question 34

query_neg_Pearson = devi_x*devi_y < 0 index_neg_Pearson = which(query_neg_Pearson) length(index_neg_Pearson)/nrow(DATASET_NAME)

Answer 34

provides in the console window the percentage of coordinates which result in negative correlations when x and y are multiplied

Question 35

query_pos_Pearson = devi_x*devi_y > 0 index_pos_Pearson = which(query_pos_Pearson) length(index_pos_Pearson)/nrow(DATASET_NAME)

Answer 35

provides in the console window the percentage of coordinates which result in positive correlations when x and y are multiplied

Question 36

x = seq(from = -10, to = 10, length.out = 100) y = x^2 plot(x, y) mean_x = mean(x) mean_y = mean(y) abline(v = mean_x, col="grey") abline(h = mean_y, col="grey")

Answer 36

generates a graph of the quadratic relationship y = x^2, bound between [-10, 10] -- while there is clearly a strong relationship here, a manually calculated Pearson coefficient would result in an answer of zero because there are as many coordinates in the positive quadrants as in the negative quadrants

Question 37

dataset_narrowed = dataset_general[, c(‘COLUMN_1’, ‘COLUMN_2’, etc.)]

Answer 37

for datasets with a crap-ton of columns, using brackets with the command "c(‘COLUMN_1’, ‘COLUMN_2’, etc.)" will reduce the number therein

Question 38

query_elim = !is.na(DATASET_NAME$BAD_COLUMN_1) & !is.na(DATASET_NAME$BAD_COLUMN_2) & DATASET_NAME$BAD_COLUMN_1 < HIGH # & DATASET_NAME$BAD_COLUMN_1 > LOW # & DATASET_NAME$BAD_COLUMN_2 < HIGH # & DATASET_NAME$BAD_COLUMN_2 > LOW # index_elim = which(query_elim) dataset_retained = dataset_narrowed[index_elim, ]

Answer 38

if the dataset in the narrowed group has bogus values, this set of commands can keep only the true values which may be more reasonable

Question 39

lowess_retained = lowess(x = dataset_retained$COLUMN_1, y = dataset_retained$COLUMN_2, f = 1/#) lines(x = lowess_retained$x, y = lowess_retained$y, col = ‘red’, lwd = 3)

Answer 39

codes to establish a lowess line for the values

Question 40

plot(x = DATASET_NAME$VARIABLE_X, y = DATASET_NAME$VARIABLE_Y, xlab = “VARIABLE X”, ylab = “VARIABLE Y”) regression_results = lm(VARIABLE_Y~VARIABLE_X, data = DATASET_NAME) lines = (x = DATASET_NAME$VARIABLE_X, y = regression_results$fitted.values)

Answer 40

plots the simple regression model between two variables, X and Y

Question 41

regression_summary = summary(regression_results)

Answer 41

establishes the summary of the simple regression for two variables

Question 42

slope = regression_summary$coefficients[“VARIABLE_X”, “Estimate”]

Answer 42

establishes the slope of the standard regression

Question 43

slope_standard_error = regression_summary$coefficients[“VARIABLE_X”, “Std. Error”]

Answer 43

establishes the standard error of the standard regression

Question 44

slope_tvalue = slope/slope_standard_error

Answer 44

calculates the t-value of the slope of the simple regression line

Question 45

intercept = regression_summary$coefficients["(Intercept)", "Estimate"] intercept_standard_error = regression_summary$coefficients["(Intercept)", "Std. Error"] intercept_tvalue = intercept/intercept_standard_error

Answer 45

code used to find the intercept t-value