SPSS LAP assignment
Danyah
stat_lab_7_sp15.pptxLAB 7:
Correlation and Linear Regression Analyses
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The main purpose of this lab is to be able to use and correctly interpret the results of the following:
Scattergram (Scatter Plot)
Correlation analysis
Simple linear regression
Scatter plot
Is used to assess (visually) the linear relationship between two quantitative variables (linear or non-linear relationship)
Is used to explore the direction of the relationship between the two variables (positive, negative or no relationship)
It can also be used to explore the data values (if there is any outliers)
Introduction
See Chapter 5- section 5.3 for more details
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Correlation Analysis:
The correlation coefficient (r) computed from the sample data measures the strength and the direction of a linear relationship between two variables.
The null hypothesis:
Ho : There is no linear relationship between the 2 variables
The alternative hypothesis:
Ha : There is a linear relationship between the 2 variables
The range of correlation coefficient is -1 to +1. When there is no linear relationship between
variables or only a weak relationship, the value of correlation coefficient will be close to 0.
No correlation: as x increases, no definite shift in y.
Positive correlation: as x increases, y increases.
Negative correlation: as x increases, y decreases
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| 0 to + 0.29: little or no association. +0.30 to + 0.49: weak positive association. + 0.5 to +0.69: medium positive association. +0.7to + 1.0: strong positive association. | 0 to - 0.29: little or no association. - 0.30 to - 0.49: weak negative association. - 0.5 to - 0.69: medium negative association. - 0.7 to - 1.0: strong negative association. |
Things to remember
Correlation coefficient cutoff points
Simple Linear Regression:
Is used to predict a single dependent variable (response) based on a single independent (predictor) variable
The null hypothesis:
Ho : The slope is zero; there is no linear relationship between the 2 variables
The alternative hypothesis:
Ha : The slope is not zero; there is a linear relationship between the 2 variables
For simple linear regression, coefficient of determination (R2) is the square of the correlation coefficient.
The range of coefficient of determination (R2) is 0 to +1. When there is no linear relationship between variables or only a weak relationship, the value of R squared will be close to 0.
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a = y intercept (constant)
b = slope (regression coefficient) of line
y = dependent (predicted )variable
x = independent (predictor) variable
y = a + bx
Y
X
Assumptions:
Normality
Equal variances
Independence
Linear relationship
Regression analysis establishes a regression equation for predictions
For a given value of x, we can predict a value of y
Correlation
Is used to measure the strength and the direction of a linear association between two quantitative variables
Simple Linear Regression
Is used to predict a single dependent variable (response) based on a single independent (predictor) variable.
Scattergram (Scatter Plot)
Is used to assess (visually) the relationship between two quantitative variables (Linear vs. non- Linear).
Hypothetical Example
For
Simple Regression
Question : Is revision time variable a good predictor of exam performance?
Answer:
Since we have two quantitative (not repeated) variables, simple linear regression should be used to answer this research question.
Step 1: Scatter plot
Dependent variable (response)? Exam performance
Independent variable (predictor)? Revision time
Can you see any relationship between revision time and exam performance? Positive or Negative?
To Obtain Scatter Plot:
From the menus choose:
Graphs Legacy Dialogs Scatter/Dot Select In the Scatterplot dialog Select the icon for simple scatter Select Define Select Exam performance as the Y-axis variable and Revision time as the X-axis variable Click Ok
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Question 1: Is revision time variable a good predictor of exam performance? Continued:
Answer:
To Obtain an Individual 95% Confidence Interval on the Scatter Plot:
After scattergram is drawn, do the following:
Double click on graph (which puts you in the chart editor window) Click on any one of the data points (this will highlight all the data points).
From the menus choose:
Click on the chart menu Select elements Fit line at total Then check individual 95% confidence interval Click apply and close properties box Close chart editor
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Can you see any relationship between Revision Time and Exam Performance? Yes, Positive relationship
Answer:
Step 2: Correlation and simple linear regression analysis
To Obtain Statistics for correlation and simple linear regression
From the menus choose:
Analyze Regression Linear Select exam performance as the dependent variable
Select revision time as the independent variable Click statistics and select estimates and 95% CI, also uncheck
Model fit and check Descriptives Click Ok
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| Model Summaryb | ||||
| Model | R | R Square | Adjusted R Square | Std. Error of the Estimate |
| 1 | .898a | .806 | .805 | 9.676 |
| a. Predictors: (Constant), Revision Time | ||||
| b. Dependent Variable: Exam Performance |
| Coefficientsa | ||||||
| Model | Unstandardized Coefficients | Standardized Coefficients | t | Sig. | ||
| B | Std. Error | Beta | ||||
| 1 | (Constant) | - 207.810 | 8.365 | -24.843 | .000 | |
| Revision Tme | 5.356 | .128 | .898 | 41.952 | .000 | |
| a. Dependent Variable: Exam Performance |
The Regression Equation:
Correlation Coefficient
Coefficient of determination (R2)
Answers the question of whether there is a significant linear relationship or not
Slope
Intercept (constant)
Revision
Time
Exam
Performance
R-square=0.806
% 80.6 of variation in exam perf. (dependent) variable explained by revision time (independent) variable
Results:
Correlation Analysis
There is a significant strong positive linear relationship between revision time and exam performance (Evidence: Correlation Coefficient (r) = 0.898)
Regression Analysis
80.6 % of the variation in exam performance is explained by revision time
Regression equation: exam performance= -207.81 + 5.36 * revision time
Conclusion: Revision time is a useful predictor of exam performance
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Exam Performance=*(Revision Time)
Exam Performance207.815.356*(Revision Ti
me)
YabX
ab
=+
+
=-+
Revision time5.360.13<0.001
R
2
= 0.806
Variableβ*
Std. ErrorP-value
Intercept = - 207.81
Dependent Variable: Exam Performance
Sheet1
| Model Summaryc | |||||||||||||
| Model | R | R Square | Adjusted R Square | Std. Error of the Estimate | Change Statistics | ||||||||
| R Square Change | F Change | df1 | df2 | Sig. F Change | |||||||||
| Basic | 1 | .339a | .115 | .104 | 9.576 | .115 | 10.316 | 3 | 238 | .000 | |||
| Smoking | 2 | .365b | .133 | .119 | 9.496 | .018 | 5.018 | 1 | 237 | .026 | 14.3153077371 | ||
| a. Predictors: (Constant), GENDER, BMI, age | -100 | ||||||||||||
| b. Predictors: (Constant), GENDER, BMI, age, EVER SMOKE CIGARETTES | -100 | ||||||||||||
| c. Dependent Variable: BASELINE DIASTOLIC BLOOD PRESSURE | -100 | ||||||||||||
| -100 | |||||||||||||
| -100 | |||||||||||||
| Model Summaryc | -100 | ||||||||||||
| Model | R | R Square | Adjusted R Square | Std. Error of the Estimate | Change Statistics | ERROR:#VALUE! | |||||||
| R Square Change | F Change | df1 | df2 | Sig. F Change | -100 | ||||||||
| 1 | .339a | .115 | .104 | 9.595 | .115 | 10.243 | 3 | 237 | .000 | -0.3339800938 | |||
| Stress | 2 | .345b | .119 | .100 | 9.613 | .004 | .559 | 2 | 235 | .573 | -3.5583878252 | ||
| a. Predictors: (Constant), GENDER, BMI, age | -100 | ||||||||||||
| b. Predictors: (Constant), GENDER, BMI, age, StresAverage, StressLow | -100 | ||||||||||||
| c. Dependent Variable: BASELINE DIASTOLIC BLOOD PRESSURE | -100 | ||||||||||||
| -100 | |||||||||||||
| -100 | |||||||||||||
| Model Summaryc | -100 | ||||||||||||
| Model | R | R Square | Adjusted R Square | Std. Error of the Estimate | Change Statistics | ERROR:#VALUE! | |||||||
| R Square Change | F Change | df1 | df2 | Sig. F Change | -100 | ||||||||
| 1 | .339a | .115 | .104 | 9.576 | .115 | 10.316 | 3 | 238 | .000 | 0 | |||
| Exercise | 2 | .381b | .145 | .123 | 9.471 | .030 | 2.769 | 3 | 235 | .042 | 18.8063330466 | ||
| a. Predictors: (Constant), GENDER, BMI, age | -100 | ||||||||||||
| b. Predictors: (Constant), GENDER, BMI, age, Exer4, Exer3, Exer2 | -100 | ||||||||||||
| c. Dependent Variable: BASELINE DIASTOLIC BLOOD PRESSURE | -100 | ||||||||||||
| -100 | |||||||||||||
| -100 | |||||||||||||
| Model Summaryc | -100 | ||||||||||||
| Model | R | R Square | Adjusted R Square | Std. Error of the Estimate | Change Statistics | ERROR:#VALUE! | |||||||
| R Square Change | F Change | df1 | df2 | Sig. F Change | -100 | ||||||||
| 1 | .339a | .115 | .104 | 9.576 | .115 | 10.316 | 3 | 238 | .000 | 0 | |||
| Cholestrol | 2 | .354b | .125 | .110 | 9.541 | .010 | 2.745 | 1 | 237 | .099 | 6.278301257 | ||
| a. Predictors: (Constant), GENDER, BMI, age | |||||||||||||
| b. Predictors: (Constant), GENDER, BMI, age, BASELINE CHOLESTEROL | |||||||||||||
| c. Dependent Variable: BASELINE DIASTOLIC BLOOD PRESSURE | |||||||||||||
| .092 | 23.9130434783 | ||||||||||||
| .114 |
Descriptive
| Summary Table - General characteristics of study participants | ||
| Characteristics | Study Participants | |
| N* | 242 | |
| Diastolic blood pressure, mm Hg a | 84.2±10.1 | |
| Body mass indexa | 30.2±6.3 | |
| Age, year a | 55.2±11.2 | |
| Cholesterol | 232.1±43.3 | |
| Gender [N, %] | ||
| Male | 83 (34.3) | |
| Female | 159 (65.7) | |
| Smoking [N, %] | ||
| Never | 157 (64.9) | |
| Ever | 85 (35.1) | |
| Self-Identified Race [N, %] | ||
| None | 99 (40.9) | |
| Mild | 75 (31.0) | |
| Moderate | 58 (24.0) | |
| Vigorous | 10 (04.1) | |
| Stress [N, %] | ||
| High | 128 (53.1) | |
| Average | 85 (35.3) | |
| Low | 28 (11.6) | |
| a Value are means± SD | ||
| * Based on the total number of subjects in the final model. |
Multiple Linear Regression
| Variable | β* | Std. Error | P-value | |
| Revision time | 5.36 | 0.13 | <0.001 | |
| Dependent Variable: Exam Performance | ||||
| Intercept = - 207.81 | ||||
| R2 = 0.806 | ||||
