Assignment-5

2B03 Assignment 5 Regression Modeling and Prediction (Chapters 10 & 11) Matthew Musulin 400329990 Due Thursday December 2 2021 Instructions: You are to use R Markdown for generating your assignment output file. You begin with the R Markdown script downloaded from A2L, and need to pay attention to information provided via introductory material posted to A2L on working with R, R Markdown, and downloading data from ODESI. Having downloaded all necessary files, placed them in the same folder/directory, and added your answers to the R Markdown script, you then are to generate your output file using “Knit to PDF” and, when complete, upload both your R Markdown file and your PDF file to the appropriate folder on A2L. 1. Define the following terms in a sentence (or short paragraph) and state a formula if appropriate (this question is worth 5 marks). i. Test of Significance A formal procedure for comparing observed data with a hypothesis we want to prove true. The hypothesis is usually a statement about the population parameters. The results of the test are expressed in terms of a probability that measures how well the data and the hypothesis agree. ii. Coefficient of Determination The measure of how well an estimated regression line fits the sample data. R 2 = SSM/SST iii. Multiple Regression Analysis When there are several explanatory variables this is known as multiple regression analysis. iv. Individual Prediction Interval An estimate of an interval in which future observations will fall, with a certain probability, given what has already been observed. Prediction intervals are often used in regression analysis. 2. An economist is studying the relationship between unemployment and inflation, and has collected the following data. Inflation appears in columns, unemployment in rows (this question is worth 5 marks). Unemployment Abated Inflation Unchanged Accelerated Total Lower 5 5 10 20 Unchanged 5 35 20 60 Higher 20 0 0 20 Total 30 40 30 100 The data in the table above summarize the relationship between unemployment and inflation based on 100 months of data. For instance, for 35 months, inflation and unemployment were unchanged, while for 10 months inflation had accelerated and unemployment was lower. 1

Using the data in the table above, conduct an appropriate hypothesis test of independence between inflation and unemployment at the 5% level of significance. row1 = c( 5 , 5 , 10 ) row2 = c( 5 , 35 , 20 ) row3 = c( 20 , 0 , 0 ) Matrix = matrix(c(row1, row2, row3), nrow= 3 , byrow= TRUE) chisq.test(Matrix, correct= TRUE) ## ## Pearson ' s Chi-squared test ## ## data: Matrix ## X-squared = 65.278, df = 4, p-value = 2.249e-13 Therefore, because the p-value 2.49e-13 < 0.05, we reject H0. There is sufficient evidence that indicates a dependence between unemployment and inflation and the result is statistically significant. 3. Consider the following dataset on the final grade received in a particular course ( grade ) and attendance ( attend , number of times present when work was handed back during the semester out of a maximum of six times). Note that R has the ability to read datafiles directly from a URL, so here (unlike the odesi data that you manually retrieved) you do not have to manually download the data providing you are connected to the internet (this question is worth 5 marks). course <- read.table( "https://socialsciences.mcmaster.ca/racinej/files/attend.RData" ) attach(course) i. Run a regression of grade on attend using the R command lm() . What is the impact of a 1 unit increase of attend on the expected grade based on your model? attach(course) ## The following objects are masked from course (pos = 3): ## ## attend, grade lm(attend~grade) ## ## Call: ## lm(formula = attend ~ grade) ## ## Coefficients: ## (Intercept) grade ## -0.53515 0.05971 Therefore, a one unit increase in attendance will increase the probability of getting a higher mark. i. In class we distinguished between correlation and causation and cautioned against inferring causation from statistical correlation. Do your results suggest that an individual who increased their attendance by 1 unit would also experience an increase in their expected grade? Why or why not? Explain the roles of confounders in this context (e.g. along the lines of Sir. R. A. Fisher’s concerns). A correlation between variables does not immediately mean that the change in one variable is the cause of the change in the values of the other variable. Causation shows that one event is the result of the occurrence of the other event. In this case, the results show that the increased attendance would also experience a higher expected grade. 2

ii. Using the PDF that accompanies the data titled "Canada Labour Statistics Division, Statistics Canada Labour Force Survey, June 2021 [Canada], Study Documentation, present a description of each variable that we select below (i.e., HRLYEARN, EDUC, SEX, AGE_12, MARSTAT, UNION ; see pages 15 and on for details). foo <- subset(lfp, FTPTMAIN== 1 & HRLYEARN > 0 , select = c(HRLYEARN,EDUC,SEX,AGE_12,MARSTAT,UNIO HRLYEARN: Usual hourly wages, employees only. EDUC: Highest educational attainment. SEX: Sex of respondent. AGE_12: Five-year age group of respondent. MARSTAT: Marital status of respondant. UNION: Union status, employees only. iii. Estimate the multivariate linear regression model and produce a model summary via summary(model) . model <- lm(HRLYEARN~EDUC+AGE_12+factor(SEX)+factor(MARSTAT)+factor(UNION), data= foo) summary(model) ## ## Call: ## lm(formula = HRLYEARN ~ EDUC + AGE_12 + factor(SEX) + factor(MARSTAT) + ## factor(UNION), data = foo) ## ## Residuals: ## Min 1Q Median 3Q Max ## -36.754 -8.336 -1.973 6.206 80.860 ## ## Coefficients: ## Estimate Std. Error t value Pr(>|t|) ## (Intercept) 18.84178 0.33855 55.655 < 2e-16 *** ## EDUC 3.65616 0.04863 75.187 < 2e-16 *** ## AGE_12 0.62894 0.02961 21.244 < 2e-16 *** ## factor(SEX)2 -4.41012 0.13528 -32.599 < 2e-16 *** ## factor(MARSTAT)2 -1.36427 0.18953 -7.198 6.22e-13 *** ## factor(MARSTAT)3 -3.36602 0.66500 -5.062 4.18e-07 *** ## factor(MARSTAT)4 -1.31162 0.43317 -3.028 0.002464 ** ## factor(MARSTAT)5 -1.16748 0.32739 -3.566 0.000363 *** ## factor(MARSTAT)6 -4.79729 0.18546 -25.867 < 2e-16 *** ## factor(UNION)2 0.43059 0.49255 0.874 0.382011 ## factor(UNION)3 -3.28675 0.14345 -22.912 < 2e-16 *** ## --- ## Signif. codes: 0 ' *** ' 0.001 ' ** ' 0.01 ' * ' 0.05 ' . ' 0.1 ' ' 1 ## ## Residual standard error: 12.65 on 36208 degrees of freedom ## Multiple R-squared: 0.2184, Adjusted R-squared: 0.2182 ## F-statistic: 1012 on 10 and 36208 DF, p-value: < 2.2e-16 What is the coefficient of determination? Interpret this result. The coefficient of determination is the measure of how well an estimated regression line fits the 4

sample data. This correlation, known as the “goodness of fit,” is represented as a value between 0.0 and 1.0. In our case, R 2 = 0 . 2184 = 21 . 84 which is low and we can say that the model is not a good fit. Interpret the coefficient estimates. Note that factor(SEX)2 is the dummy variable that represents the difference between wages between group 2 and the base group group 1, other things equal (i.e., 2=Female, 1=Male). Also, note that AGE_12 contains five year ranges, so the coefficient is the change in expected wages for an additional five years of age, other things equal. HRLYEARN is the dependent variable and EDUC, AGE_12, SEX, MARSTAT, and UNION are independent variables. Independent Variable Interpretation: The estimated value of EDUC is 3.65616. This means, with the increase in 1% of EDUC, HRLYEARN will increase by 3.65616% The estimated value of AGE_12 is 0.62894. This means, with the increase in 1% of AGE_12, HRLYEARN will increase by 0.62894%. The estimated value of factor(SEX)2 is -4.41012. This means, with the increase in 1% of fac- tor(SEX)2, HRLYEARN will decrease by -4.41012%. The estimated value of factor(MARSTAT)2 is -1.36427. This means, with the increase in 1% of (MARSTAT)2, HRLYEARN will decrease by 1.36427%. The estimated value of factor factor(MARSTAT)3 is -3.36602. This means, with the increase in 1% of (MARSTAT)3, HRLYEARN will decrease by 3.36602%. The estimated value of factor(MARSTAT)4 is -1.31162. This means, with the increase in 1% of (MARSTAT)4, HRLYEARN will decrease by 1.31162%. The estimated value of factor(MARSTAT)5 is -1.16748. This means, with the increase in 1% of factor(MARSTAT)5, HRLYEARN will decrease by 1.16748%. The estimated value of factor(MARSTAT)6 is -4.79729. This means, with the increase in 1% of factor(MARSTAT)6, HRLYEARN will decrease by 4.79729%. The estimated value of factor(UNION)2 is 0.43059. This means, with the increase in 1% of factor(UNION)2, HRLYEARN will increase by 0.43059%. The estimated value of factor(UNION)3 is -3.28675. This means, with the increase in 1% of factor(UNION)3, HRLYEARN will decrease by 3.28675%. What is the impact of either being covered but not unionized or being non-unionized relative to unionized workers? Are they both significant? Does this make sense? Significance of factor(UNION)2: H0 = (UNION)2 = 0 HA = (UNION)2 =/= 0 The p-value of factor (UNION)2 is 0.382011. Therefore we accept H0. This means factor(UNION)2 is non-significant. factor(UNION)2 has no effect on the dependent variable HRLYEARN. Significance of factor(UNION)3: H0 = (UNION)3 = 0 HA = (UNION)3 =/= 0 The p-value of factor (UNION)3 is 2e-16. Therefore we reject H0. This means factor(UNION)3 is significant. factor(UNION)3 has a negative effect on the dependent variable HRLYEARN because the coefficient value is -3.36436. Conduct a test of significance for the variable EDUC . Show all steps and interpret the results. Significance of EDUC: H0 = EDUC = 0 HA = EDUC =/= 0 The p-value of EDUC is 2e-16. Therefore we reject H0. This means EDUC is significant. EDUC has a positive effect on the dependent variable HRLYEARN because the coefficient value is 3.45415. 5