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Multivariate Regression Equation What is the difference between univariate an...
Sep 5, 2024
Multivariate Regression Equation What is the difference between univariate and multivariate regression? Write the univariate explanatory regression equation in general form. How does the univariate regression equation look like in matrix form? (█(y1@y2@y3@⋮@yn)) = (█( 1 x11 @1 x21@1 x31@⋮@1 xn1)█(x12@x22@x32@⋮@xn2)█( ⋯@ ⋯@ ⋯@ ⋮@ ⋯)█( x1m@ x2m@ x3m@ ⋮@ xnm)) (█(b0@b1@b2@bm)) + (█(e1@e2@e3@⋮@en)) 2a) What does the y represent? 2b) what does 1 represent? 2c) what do the xs represent? 2d) what do bo and b1 through bm represent? 2e) What does e represent? How do we extend the univariate equation in matrix form to the multivariate equation for 2 IVs or predictors and 2 DVs? (█(y11@y21@y31@⋮@yn1)█( y12@ y22@ y32@⋮@ yn2)) = (█(1 x11@1 x21@1 x31@⋮@ 1 xn1)█( x12@ x22@ x32@⋮@xn2)) (█(b01@b11@b21)█( b02@ b12@ b22)) + (█(e11@e21@e31@⋮@en1)█( e12@ e22@ e32@ ⋮@ en2)) 3a) What do the ys represent? 3b) What do the xs represent? 3c) What do the b0s, b1s, and b2s represent? 3d) What do the e1s and e2s represent? How can we write the prediction equation for person 1 for y1? How can we write the prediction equation for person 1 for y2? To indicate the multivariate regression equation textbooks use different Greek letters to represent the model parameters, some use the uppercase version of the univariate equation and most bold the equation. To keep it as simple as possible, for this class we use upper case letters and bold the equation Y = B0 + BX + E Note, in the MLR the model parameters represent matrices reflecting the number of dependent and independent variables. Diagramming the MLR Models Multivariate linear regression equations and their matrix notation may not be as easily understood or intuitive. One way to represent statistical models are through diagrams. Observed variables are represented by rectangular boxes Causal relationships are represented by directed edges Correlations/covariances are represented by bi-directional edges Intercepts are represented by triangles (often they are left out of diagrams for simplicity) Diagram the multivariate regression equation using Model 6 from Handout 1. Students spelling and arithmetic scores are predicted by SES and stability. Label each observed variable and include all regression coefficients in the diagram! Checking Assumptions in Multivariate Linear Regression Every statistical method has assumptions. Assumptions mean that your data must satisfy certain properties for statistical results to be accurate. Violations may affect the accuracy and or robustness of results. Let’s review the assumptions underlying univariate linear regression. What are key assumptions and how do we check them? Normally distributed DV – Linear relationship between IVs (predictors) and DV (outcome) No outliers – independent observations no multicollinearity of IVs - Normally distributed residuals – homoscedasticity of residuals (equal variances) – The assumptions for Multivariate Multiple Linear Regression are essentially the same but the normality assumption now extends to multivariate normality: DVs follow a multivariate normal distribution – Linear relationship between IVs (predictors) and DVs (outcome), No Outliers – Independent observations Similar spread across range – residuals – multivariate normal – No Multicollinearity – When to use Multivariate Multiple Linear Regression? You should use Multivariate Multiple Linear Regression in the following scenario: You want to predict multiple continuous outcomes from the same unit of observation (measured at a single occasion) from one or more IVs or predictor variables (both continuous and categorical) You have no repeated measures from the same unit of observation over multiple occasions.
Solution by Steps
step 1
The difference between univariate and multivariate regression is that univariate regression involves one dependent variable and one independent variable, while multivariate regression involves multiple dependent variables and one or more independent variables
step 2
The univariate explanatory regression equation in general form is given by: y=b0+b1x+ey = b_0 + b_1x + e, where yy is the dependent variable, b0b_0 is the intercept, b1b_1 is the coefficient of the independent variable xx, and ee is the error term
step 3
The univariate regression equation in matrix form can be represented as: (y1y2y3⋮yn)=(1amp;x11amp;x12amp;⋯amp;x1m1amp;x21amp;x22amp;⋯amp;x2m1amp;x31amp;x32amp;⋯amp;x3m⋮amp;⋮amp;⋮amp;⋮1amp;xn1amp;xn2amp;⋯amp;xnm)(b0b1b2⋮bm)+(e1e2e3⋮en) \begin{pmatrix} y_1 \\ y_2 \\ y_3 \\ \vdots \\ y_n \end{pmatrix} = \begin{pmatrix} 1 & x_{11} & x_{12} & \cdots & x_{1m} \\ 1 & x_{21} & x_{22} & \cdots & x_{2m} \\ 1 & x_{31} & x_{32} & \cdots & x_{3m} \\ \vdots & \vdots & \vdots & \vdots \\ 1 & x_{n1} & x_{n2} & \cdots & x_{nm} \end{pmatrix} \begin{pmatrix} b_0 \\ b_1 \\ b_2 \\ \vdots \\ b_m \end{pmatrix} + \begin{pmatrix} e_1 \\ e_2 \\ e_3 \\ \vdots \\ e_n \end{pmatrix}
step 4
In the multivariate regression equation for 2 independent variables (IVs) or predictors and 2 dependent variables (DVs), the equation can be expressed as: (y11amp;y12y21amp;y22y31amp;y32⋮amp;⋮yn1amp;yn2)=(1amp;x11amp;x121amp;x21amp;x221amp;x31amp;x32⋮amp;⋮1amp;xn1amp;xn2)(b01amp;b02b11amp;b12b21amp;b22)+(e11amp;e12e21amp;e22e31amp;e32⋮amp;⋮en1amp;en2) \begin{pmatrix} y_{11} & y_{12} \\ y_{21} & y_{22} \\ y_{31} & y_{32} \\ \vdots & \vdots \\ y_{n1} & y_{n2} \end{pmatrix} = \begin{pmatrix} 1 & x_{11} & x_{12} \\ 1 & x_{21} & x_{22} \\ 1 & x_{31} & x_{32} \\ \vdots & \vdots \\ 1 & x_{n1} & x_{n2} \end{pmatrix} \begin{pmatrix} b_{01} & b_{02} \\ b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} + \begin{pmatrix} e_{11} & e_{12} \\ e_{21} & e_{22} \\ e_{31} & e_{32} \\ \vdots & \vdots \\ e_{n1} & e_{n2} \end{pmatrix}
step 5
In the prediction equation for person 1 for y1y_1, it can be written as: y1=b0+b1x11+b2x12+e1 y_1 = b_0 + b_1x_{11} + b_2x_{12} + e_1
step 6
For person 1 for y2y_2, the prediction equation is: y2=b0+b1x21+b2x22+e2 y_2 = b_0 + b_1x_{21} + b_2x_{22} + e_2
Answer
The differences between univariate and multivariate regression, the general form of the univariate regression equation, its matrix representation, the extension to multivariate regression, and the prediction equations for specific individuals are all outlined above.
Key Concept
Understanding the structure and differences between univariate and multivariate regression is crucial for statistical modeling.
Explanation
The answers provided clarify the fundamental concepts of regression analysis, including the equations and their representations, which are essential for predicting outcomes based on multiple variables.
Scenario: To improve the college experience for undergraduate students, a southwestern university in the United States collected data from 87 undergraduates on students’ perception of academic workload, academic pressure, financial concerns, and the quality of social relationships on campus and the frequency with which they experience depression, anxiety, feelings of isolation, and insecurity about the future. The institutional research unit wanted to know whether students perceived academic workload, their perceived academic pressure, and the quality of social relationships predicted jointly experienced depression and anxiety. All variables were measured on a scale from 1 to 7. 在这个场景中，大学正在研究多种因素（例如学术工作量、学术压力、财务问题、社交关系的质量）对学生心理健康（如抑郁和焦虑）的影响。研究的目的是了解这些变量如何预测学生共同经历的抑郁和焦虑。 多元线性回归的典型问题 NOTE: Use alpha of .05 for all statistical significance questions! 1) Set your work directory wherever you saved your data on your computer. Paste your code below. (2 points) setwd("/Users/wq666888/Desktop/博三1nd课程资料/EDP646_a Multivariate Method/第二周") 2) Import the StudentMentalHealth.csv data set and store it in an object. Paste the code you used below. (4 points) data <- read.csv("StudentsMentalHealthSurvey.csv") 3) Look at descriptive statistics for the data. Paste your code below (1 point). Is there any missing data, or do we have 87 responses on all variables? (1 point) summary(data) sum(is.na(data)) [1] 0 Length:87 The result is 0, meaning there are no missing values, and I have 87 responses for all variables. 4) Run a multivariate regression analysis, jointly predicting student depression and anxiety from perceived academic workload, perceived academic pressure, and perceived quality of social relationships. Paste your code and output using the summary() below (5 points). model <- lm(cbind(depression, anxiety) ~ academic_workload + academic_pressure + social_relationships, data = data) summary(model) Response depression : Call: lm(formula = depression ~ academic_workload + academic_pressure + social_relationships, data = data) Residuals: Min 1Q Median 3Q Max -2.93867 -0.85819 0.07506 0.78568 2.45842 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.3915 0.7055 1.972 0.05189 . academic_workload 0.3796 0.1661 2.285 0.02485 * academic_pressure 0.3417 0.1260 2.712 0.00813 ** social_relationships -0.3379 0.1090 -3.101 0.00264 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1.181 on 83 degrees of freedom Multiple R-squared: 0.2803, Adjusted R-squared: 0.2543 F-statistic: 10.78 on 3 and 83 DF, p-value: 4.708e-06 Response anxiety : Call: lm(formula = anxiety ~ academic_workload + academic_pressure + social_relationships, data = data) Residuals: Min 1Q Median 3Q Max -2.9827 -0.8601 0.1263 0.8212 2.4051 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.7994 0.6690 2.690 0.008645 ** academic_workload 0.4058 0.1575 2.576 0.011763 * academic_pressure 0.2336 0.1195 1.955 0.053902 . social_relationships -0.3742 0.1033 -3.621 0.000503 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 1.12 on 83 degrees of freedom Multiple R-squared: 0.2813, Adjusted R-squared: 0.2554 F-statistic: 10.83 on 3 and 83 DF, p-value: 4.448e-06 5) Look at the separate regression results. Which predictors are statistically significant in predicting anxiety? (2 points). The significance of predictors is determined based on the p-value from the regression coefficients. When the p-value is less than 0.05, the predictor is considered statistically significant. According to the data analysis results, the p-value for academic_workload is 0.011763, which is significant (p < 0.05); the p-value for social_relationships is 0.000503, which is highly significant (p < 0.001); and the p-value for academic_pressure is 0.053902, slightly greater than 0.05, indicating that its effect is statistically near significant but not fully significant. 6) Interpret the regression coefficient for social relationships. (2 points). In the regression model predicting anxiety, the coefficient for social_relationships is -0.3742. This means that for each one-unit increase in the social relationships score, the anxiety score is expected to decrease by 0.3742, holding all other variables constant. Because the coefficient is negative, this indicates an inverse relationship between social relationships and anxiety. As the quality of social relationships improves, the level of anxiety decreases. In the regression model predicting depression, the coefficient for social_relationships is -0.3379. This means that for each one-unit increase in the social relationships score, the depression score is expected to decrease by 0.3379, holding all other variables constant. Because the coefficient is negative, this indicates an inverse relationship between social relationships and depression. As the quality of social relationships improves, the level of depression decreases. 7) Interpret the regression coefficient for study satisfaction. (2 points). 8) How much variance in anxiety scores is explained by the model? (2 points). 9) Now take a look at the results regression predicting depression. Which predictors are statistically significant in predicting anxiety? (2 points). 10) Interpret the regression coefficient for academic workload. (2 points). 11) How do we interpret the Residual Standard Error in this depression model? How do evaluate the size of the error? (3 points) 12) Perform the multivariate test to evaluate the statistical significance f the joint effect of each predictor on the combination of depression and anxiety using the summary() and Manova(). Paste your code below (5 points). 13) Considering results from the Wilks and Pillai multivariate tests, which predictor variables have a statistically significant joint effect on depression and anxiety? (2 points). 14) How do you make sense of the results from the multivariate tests and the univariate tests from the separate analyses (2 points) and what should the final model be based on our results (2 points)? BONUS: You do not have to complete the questions below. Skipping the questions will not hurt your grade. However, any points you earn on the questions below can help make up for points missed elsewhere. 15) Check for multivariate normality of the residuals. Are they distributed normally? (5 points). 16) Conduct a model comparison to evaluate model fit of the original model and your final model. Which model fits the data better and why? (3points).
Solution by Steps
step 1
Set your working directory to the location where the data is saved. Use the following code:
setwd(&quot;/Users/wq666888/Desktop/博三1nd课程资料/EDP646_a Multivariate Method/第二周&quot;)
step 2
Import the dataset "StudentsMentalHealthSurvey.csv" and store it in an object called data. Use the following code:
data &lt;- read.csv(&quot;StudentsMentalHealthSurvey.csv&quot;)
step 3
Check the descriptive statistics for the data and look for missing values. Use the following code:
summary(data) \quad \text{and} \quad sum(is.na(data))
step 4
The result of sum(is.na(data)) is 0, indicating there are no missing values, and the length of the data is 87, confirming 87 responses for all variables
step 5
Run a multivariate regression analysis to predict student depression and anxiety from perceived academic workload, academic pressure, and social relationships. Use the following code:
model &lt;- lm(cbind(depression, anxiety) ~ academic_workload + academic_pressure + social_relationships, data = data)
step 6
Use the summary function to view the results of the regression analysis:
summary(model)
step 7
Identify statistically significant predictors for anxiety based on p-values. The p-values for academic_workload (0.011763) and social_relationships (0.000503) are significant, while academic_pressure (0.053902) is near significant
step 8
Interpret the regression coefficient for social_relationships. A coefficient of -0.3742 indicates that for each one-unit increase in social relationships, anxiety decreases by 0.3742, holding other variables constant
step 9
The coefficient for social_relationships in the depression model is -0.3379, indicating a similar inverse relationship with depression
step 10
The variance in anxiety scores explained by the model can be assessed using the R-squared value from the summary output
step 11
Review the regression results for depression to identify significant predictors. The significant predictors are academic_workload and social_relationships
step 12
Interpret the regression coefficient for academic_workload, which indicates the expected change in anxiety for each unit increase in academic workload
step 13
The Residual Standard Error in the depression model indicates the average distance that the observed values fall from the regression line. A smaller value indicates a better fit
step 14
Perform a multivariate test using the Manova function to evaluate the joint effect of predictors on depression and anxiety. Use the following code:
manova_model &lt;- manova(cbind(depression, anxiety) ~ academic_workload + academic_pressure + social_relationships, data = data)
step 15
Analyze the results from the Wilks and Pillai multivariate tests to determine which predictors have a statistically significant joint effect on depression and anxiety
step 16
Interpret the results from both multivariate and univariate tests to finalize the model based on statistical significance and practical relevance
Answer
The analysis indicates that academic workload and social relationships are significant predictors of both anxiety and depression, while academic pressure is near significant. The model explains a portion of the variance in anxiety and depression scores.
Key Concept
Multivariate regression analysis allows for the simultaneous examination of multiple predictors on multiple outcomes.
Explanation
The results show how academic workload and social relationships significantly impact students' mental health, providing insights for interventions.
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