Statistics II · Unit 4 · 6 hrs
Multiple correlation and regression
Exam-focused notes for Multiple correlation and regression (Statistics II, STA215): what the TU syllabus asks and how it has actually been tested, with 13 solved past questions from this unit.
What this unit covers
- Multiple and partial correlation
- Introduction of multiple linear regression
- Hypothesis testing of multiple regression
- Test of significance of regression
- Test of individual regression coefficient
- Model adequacy tests
- Problems and illustrative examples related to computer Science and IT
Introduction of multiple linear regression
Multiple Regression Analysis
Multiple regression analysis is applied to: - Establish a functional relationship between one dependent variable $Y$ and two or more independent variables $X1, X2, \ldots$ - Predict/estimate the value of the dependent variable from known independent variabl...
Full solved answer →Multiple Regression Analysis: IRS Unpaid Taxes Estimation
For the multiple regression model $Y = \beta0 + \beta1 X1 + \beta2 X2 + \epsilon$, the error term $\epsilon$ must satisfy: 1. Zero mean: $E(\epsiloni) = 0$ for all $i$. 2. Constant variance (homoscedasticity): $Var(\epsiloni) = \sigma^2$ for all $i$. 3. Ind...
Full solved answer →Multiple Linear Regression Analysis for Computer Vision Syndrome
ID CVS (Y) Age X₁ Work hrs X₂ ---------------------------------- 001 6 24 4 007 7 26 5 125 5 30 6 231 11 41 8 99 3 47 3 299 29.0 50 6 145 28 52 7 $n = 7$. CVS measured on scale 0 to 50. Dependent variable: Scale of CVS ($Y$) - it is the outcome the study wa...
Full solved answer →Multiple Linear Regression Analysis
Obs Lifetime (years) Play time (hrs/day) RAM (MB) ------------------------------------------------------ 1 5 2 8 2 1 8 2 3 7 1 6 4 2 5 3 5 3 6 2 6 4 3 4 7 6 2 7 $n = 7$. Dependent variable: Lifetime (Y), since it is being affected/explained. Independent var...
Full solved answer →From the following information of variables X₁, X₂, and Y:
- $\Sigma X_1 = 272$
- $\Sigma X_2 = 441$
- $\Sigma Y = 147$
- $\Sigma X_1^2 = 7428$
- $\Sigma X_2^2 = 19461$
- $\Sigma Y^2 = 2173$
- $\Sigma X_1 Y = 4013$
- $\Sigma X_1 X_2 = 12005$
- $\Sigma X_2 Y = 6485$
- $n = 10$
Fit a regression equation Y on X₁ and X₂. Interpret the regression coefficients. [10]
Multiple Linear Regression (MLR) is a statistical technique that models the relationship between one dependent variable and two or more independent variables. It allows us to estimate/predict the dependent variable using several predictors simultaneously. G...
Full solved answer →Multiple and partial correlation
Explain the concept of multiple and partial correlation coefficients. Consider three variables X1, X2 and X3. If $r_{12}=0.40$, $r_{23}=0.50$ and $r_{13}=0.6$ find $R_{123}$ and $r_{23.1}$. [5]
Three variables $X1, X2, X3$ with zero-order correlation coefficients: $$r{12} = 0.40, \quad r{23} = 0.50, \quad r{13} = 0.60$$ Note on notation: The problem statement contains a typo. "$R{123}$" is standardly interpreted as the multiple correlation coeffic...
Full solved answer →Define multiple correlation. In a trivariate distribution X1, X2, and X3, the simple correlation coefficients are given as r12= 0.5, r23=0.6 and r13=0.7. Find i. partial correlation coefficient between X1 and X2 keeping X3 constant. ii. multiple correlation coefficient assuming X1 as dependent variable. [5]
Multiple correlation is the correlation between one variable (the dependent variable) and the combined linear effect of two or more other variables (independent variables) taken together. For a trivariate distribution $X1, X2, X3$, the multiple correlation ...
Full solved answer →Write short notes on the following. i. Partial and multiple correlation coefficient. ii. Properties of good estimator. [5]
--- Definition: The partial correlation coefficient measures the degree of linear relationship between two variables while eliminating (holding constant) the effect of one or more other variables. For three variables X1, X2, and X3, the partial correlation ...
Full solved answer →Test of significance of regression
Multiple Regression Analysis Question
A multiple regression equation yields the following results:
| Source | Sum of square | Degree of freedom |
|---|---|---|
| Regression | 740 | 2 |
| Error | 510 | 17 |
i) What is the total sample size?
ii) How many independent variables are being considered?
iii) Compute the coefficient of determination and interpret its value.
iv) Compute the standard error of estimate.
v) Test the hypothesis that the overall fit of the model is significant or not. Assume $\alpha=0.05$.
[5]
Source Sum of Squares (SS) Degrees of Freedom (df) --------------------------------------------------- Regression 740 2 Error 510 17 Total 1250 19 $\alpha = 0.05$ --- $$df{Total} = df{Reg} + df{Error} = 2 + 17 = 19$$ Since $df{Total} = n - 1$: $$n - 1 = 19 ...
Full solved answer →Regression Analysis: Computer Program Efficiency
Given regression equation: $$Y = 52.7 - 2.87X_1 + 0.85X_2$$
Where:
- Y = Processed requests per hour
- X₁ = Data size (gigabytes)
- X₂ = Number of tables
- Total sum of squares (TSS) = 1452
- Sum of squares due to regression (SSR) = 1143.3
- Standard error of b₂ = 0.55
- n = 7 observations
- Regression equation: $\hat{Y} = 52.7 - 2.87X1 + 0.85X2$ - Total Sum of Squares: $SST = 1452$ - Sum of Squares due to Regression: $SSR = 1143.3$ - Standard error of $b2$: $S{b2} = 0.55$ - Sample size: $n = 7$; number of predictors: $k = 2$ - Significance l...
Full solved answer →ANOVA Analysis for Multiple Regression Model
Source SS df MSS F --------------- Regression 12.62 2 ? ? Error 0.78 12 ? Total 13.40 14 - Number of independent variables: $k = 2$ - Total $df = n - 1 = 14 \Rightarrow n = 15$ --- MSS due to Regression: $$MSR = \frac{SSR}{dfR} = \frac{12.62}{2} = 6.31$$ MS...
Full solved answer →Suppose we are given following information with n=7, multiple regression model is $\hat{A} = 8.15 + 0.6 X_1 + 0.54 X_2$. Here, Total sum of square = 1493, and Sum of square due to error = 91. Find i) $R^2$ and interpret it. ii) Test the overall significance of model [5]
- Sample size: $n = 7$ - Model: $\hat{A} = 8.15 + 0.6X1 + 0.54X2$ - Number of independent variables: $k = 2$ - Total Sum of Squares: $SST = 1493$ - Sum of Squares due to Error: $SSE = 91$ --- Sum of Squares due to Regression: $$SSR = SST - SSE = 1493 - 91 =...
Full solved answer →Model adequacy tests
Write shorts notes on the following: i. Test of equality of two variances ii. Adjusted R2R^2R2 [5]
The test of equality of two variances (also called the F-test) is used to determine whether two population variances are equal. It is based on the F-distribution and is a parametric test. - Null Hypothesis (H₀): σ₁² = σ₂² (The two population variances are e...
Full solved answer →Make Unit 4 stick
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