This document provides an introduction to linear regression analysis. It discusses how regression finds the best fitting straight line to describe the relationship between two variables. The regression line minimizes the residuals, or errors, between the predicted Y values from the line and the actual data points. The accuracy of predictions from the regression model can be evaluated using the correlation coefficient (r) and the standard error of estimate. Multiple linear regression extends this process to model relationships between a dependent variable Y and two or more independent variables (X1, X2, etc).
The document discusses regression and correlation analysis between BMI (Kg/m2) of pregnant mothers and birth weight (kg) of their newborns using data from 15 mothers. A scatter plot showed a positive linear relationship between BMI and birth weight. Linear regression was used to calculate the regression line as y=1.775351+0.0330817x, which can be used to predict birth weight based on a mother's BMI. The correlation coefficient (R) between BMI and birth weight was 0.94, indicating a strong positive correlation.
Linear regression and correlation analysis ppt @ bec domsBabasab Patil
This document introduces linear regression and correlation analysis. It discusses calculating and interpreting the correlation coefficient and linear regression equation to determine the relationship between two variables. It covers scatter plots, the assumptions of regression analysis, and using regression to predict and describe relationships in data. Key terms introduced include the correlation coefficient, linear regression model, explained and unexplained variation, and the coefficient of determination.
This document provides an overview of regression analysis, including linear regression, multiple regression, and assessing assumptions. It defines regression as a technique for investigating relationships between variables. Simple linear regression involves one predictor and one response variable, while multiple regression extends this to multiple predictors. Key steps are outlined such as assessing the fit of regression models using R-squared, testing the significance of individual predictors, and ensuring assumptions of normality, linearity and equal variance are met. Examples are provided demonstrating how to evaluate these assumptions and interpret regression results.
The document discusses correlation and linear regression. It defines Pearson and Spearman correlation as statistical techniques to measure the relationship between two variables. Pearson correlation measures the linear association between interval variables, while Spearman correlation measures statistical dependence between two variables using their rank order. Linear regression finds the best fit linear relationship between a dependent and independent variable to predict changes in one based on the other. The key assumptions and interpretations of correlation coefficients and regression lines are also covered.
1. The document discusses correlation and correlation coefficients, which measure the strength and direction of association between two variables.
2. A correlation coefficient ranges from 0, indicating no correlation, to 1 or -1, indicating perfect positive or negative correlation. Coefficients above 0.5 generally indicate a strong linear relationship.
3. The Pearson correlation coefficient (r) specifically measures the linear correlation between two normally distributed variables, while the Spearman correlation (rs) is nonparametric and assesses correlation between ordinal or non-normally distributed variables.
4. Correlation only indicates association, not causation. Significant correlation is also not necessarily clinically meaningful. Correlation coefficients and their statistical significance must be interpreted carefully.
Chapter 16: Correlation
(enhanced by VisualBee)nunngera
Correlation is a statistical method used to measure the relationship between two variables. A relationship exists when changes in one variable are accompanied by consistent changes in the other. A correlation evaluates the direction, form, and degree of the relationship. The Pearson correlation specifically measures the direction and strength of a linear relationship between two numerical variables. Other correlational methods like Spearman and point-biserial correlations can be used for ordinal or dichotomous variable relationships.
This document provides an overview of linear regression analysis. It defines key terms like dependent and independent variables. It describes simple linear regression, which involves predicting a dependent variable based on a single independent variable. It covers techniques for linear regression including least squares estimation to calculate the slope and intercept of the regression line, the coefficient of determination (R2) to evaluate the model fit, and assumptions like independence and homoscedasticity of residuals. Hypothesis testing methods for the slope and correlation coefficient using the t-test and F-test are also summarized.
The document discusses correlation, regression analysis, and an example analysis. It defines correlation as a measure of the strength of association between two variables. Regression analysis establishes a mathematical relationship between variables to predict outcomes. The example analyzes the correlation between residents' duration of residence in a city and their attitude toward the city, finding a strong positive correlation. It then performs a bivariate regression to model this relationship mathematically.
This presentation describes the application of regression analysis in research, testing assumptions involved in it and understanding the outputs generated in the analysis.
This document provides an overview of correlation analysis procedures in SPSS, including bivariate correlation, partial correlation, and distance measures. It discusses interpreting correlation coefficients and significance values. Scatterplots are recommended to check assumptions before correlation. Hands-on exercises are included to find correlations between variables while controlling for other variables.
This document provides an overview of correlation and the Pearson correlation coefficient. It discusses how the Pearson r describes the direction, form, and strength of the linear relationship between two variables. It explains how to calculate r using the sum of products formula and interpret the results. The text also covers hypothesis testing with r and reporting correlations. Alternatives to the Pearson r are mentioned but not covered in detail.
Correlation describes the relationship between two or more variables. A positive correlation means that as one variable increases, the other also increases, while a negative correlation means that as one variable increases, the other decreases. Correlation is measured numerically using coefficients like the Pearson correlation coefficient r, which ranges from -1 to 1, with values farther from 0 indicating stronger linear relationships and the direction indicating positive or negative correlation. Correlation is used in business and economics to study relationships between variables like price and demand.
Regression analysis is a statistical technique used to model relationships between variables. It allows one to predict the average value of a dependent variable based on the value of one or more independent variables. The key ideas are that the dependent variable is influenced by the independent variables in a linear or curvilinear fashion, and regression provides an equation to estimate the dependent variable given values of the independent variables. Common applications of linear regression include forecasting, determining relationships between variables, and estimating how changes in one variable impact another.
Simple Linear Regression is a statistical technique that attempts to explore the relationship between one independent variable (X) and one dependent variable (Y). The Simple Linear Regression technique is not suitable for datasets where more than one variable/predictor exists.
This document discusses correlation and linear regression. It defines correlation as the association between two variables, which can be positive, negative, or non-existent. Linear correlation exists when plotted points approximate a straight line. The correlation coefficient r measures the strength of a linear relationship between -1 and 1. Linear regression finds the linear relationship that best fits the data using a regression equation to predict y values from x. Multiple linear regression extends this to use multiple explanatory variables.
This document discusses correlation and linear regression analysis. It begins by outlining the learning objectives which are to describe relationships between variables using correlation, estimate effects of independent variables on dependents with regression, and perform and interpret different types of regression analyses. It then provides examples of how correlation calculates the strength and direction of relationships between interval variables and how regression finds the best fitting linear equation to estimate relationships between variables. It emphasizes that regression minimizes the sum of squared errors to find the line of best fit for the data.
Correlation and regression analysis are statistical tools used to analyze relationships between variables. Correlation measures the strength and direction of association between two variables on a scale from -1 to 1. Regression analysis uses one variable to predict the value of another variable and draws a best-fit line to represent their relationship. There are always two lines of regression - one showing the regression of x on y and the other showing the regression of y on x. Regression coefficients from these lines indicate the slope and intercept of the lines and can help estimate unknown variable values based on known values.
Correlation analysis is a statistical technique used to determine the degree of relationship between two quantitative variables. Scatterplots are used to graphically depict the relationship and identify if it is positive, negative, or no correlation. The correlation coefficient measures the strength and direction of correlation, ranging from -1 to 1. A significance test determines if a correlation is likely to have occurred by chance or is statistically significant. Different types of correlation include simple, multiple, partial, and autocorrelation.
Regression analysis allows researchers to identify an equation that best fits paired data and predict the relationship between two quantitative variables. Linear regression assumes a linear relationship and finds the line that best describes how the dependent variable changes with the independent variable. The regression line equation takes the form Y = bX + a, where b is the slope and a is the intercept. Researchers can use linear regression to predict new Y values based on X and assess how well the linear model fits the data.
This document provides an overview of correlation and linear regression analysis. It defines correlation as a statistical measure of the relationship between two variables. Pearson's correlation coefficient (r) ranges from -1 to 1, with values farther from 0 indicating a stronger linear relationship. Positive values indicate an increasing relationship, while negative values indicate a decreasing relationship. The coefficient of determination (r2) represents the proportion of shared variance between variables. While correlation indicates linear association, it does not imply causation. Multiple regression allows predicting a continuous dependent variable from two or more independent variables.
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This document discusses methods for analyzing the relationship between two quantitative variables, including:
- Scatter diagrams can show the relationship and be used to identify if the variables are positively or negatively correlated.
- The linear correlation coefficient, r, quantifies the strength of the linear relationship between -1 and 1, where values closer to -1 or 1 indicate a stronger negative or positive correlation, respectively.
- Least-squares regression finds the best-fitting straight line to describe the linear relationship between two variables by minimizing the sum of the squared residuals. It can be used to make predictions, but may not be accurate far outside the original data range.
This document discusses methods for analyzing the relationship between two quantitative variables, including:
- Scatter diagrams can show the relationship and be used to identify if the variables are positively or negatively correlated.
- The linear correlation coefficient, r, quantifies the strength of the linear relationship between -1 and 1, with values closer to 1 or -1 indicating a stronger linear relationship.
- Least-squares regression finds the best-fitting straight line to describe the linear relationship between two variables by minimizing the sum of the squared residuals. It can be used to make predictions, but may not be accurate far outside the original data range.
Assessment 2 ContextIn many data analyses, it is desirable.docxfestockton
Assessment 2 Context
In many data analyses, it is desirable to compute a coefficient of association. Coefficients of association are quantitative measures of the amount of relationship between two variables. Ultimately, most techniques can be reduced to a coefficient of association and expressed as the amount of relationship between the variables in the analysis. There are many types of coefficients of association. They express the mathematical association in different ways, usually based on assumptions about the data. The most common coefficient of association you will encounter is the Pearson product-moment correlation coefficient (symbolized as the italicized r), and it is the only coefficient of association that can safely be referred to as simply the "correlation coefficient". It is common enough so that if no other information is provided, it is reasonable to assume that is what is meant.
Correlation coefficients are numbers that give information about the strength of relationship between two variables, such as two different test scores from a sample of participants. The coefficient ranges from -1 through +1. Coefficients between 0 and +1 indicate a positive relationship between the two scores, such as high scores on one test tending to come from people with high scores on the second. The other possible relationship, which is every bit as useful, is a negative correlation between -1 and 0. A negative correlation possesses no less predictive power between the two scores. The difference is that high scores on one measure are associated with low scores on the other.
An example of the kinds of measures that might correlate negatively is absences and grades. People with higher absences will be expected to have lower grades. When a correlation is said to be significant, it can be shown that the correlation is significantly different form zero in the population. A correlation of zero means no relationship between variables. A correlation other than zero means the variables are related. As the coefficient gets further from zero (toward +1 or -1), the relationship becomes stronger.Interpreting Correlation: Magnitude and Sign
Interpreting a Pearson's correlation coefficient (rXY) requires an understanding of two concepts:
· Magnitude.
· Sign (+/-).
The magnitude refers to the strength of the linear relationship between Variable X and Variable
The rXY ranges in values from -1.00 to +1.00. To determine magnitude, ignore the sign of the correlation, and the absolute value of rXY indicates the extent to which Variable X and Variable Y are linearly related. For correlations close to 0, there is no linear relationship. As the correlation approaches either -1.00 or +1.00, the magnitude of the correlation increases. Therefore, for example, the magnitude of r = -.65 is greater than the magnitude of r = +.25 (|.65| > |.25|).
In contrast to magnitude, the sign of a non-zero correlation is either negative or positive.
These labels are not interpreted ...
Assessment 2 ContextIn many data analyses, it is desirable.docxgalerussel59292
Assessment 2 Context
In many data analyses, it is desirable to compute a coefficient of association. Coefficients of association are quantitative measures of the amount of relationship between two variables. Ultimately, most techniques can be reduced to a coefficient of association and expressed as the amount of relationship between the variables in the analysis. There are many types of coefficients of association. They express the mathematical association in different ways, usually based on assumptions about the data. The most common coefficient of association you will encounter is the Pearson product-moment correlation coefficient (symbolized as the italicized r), and it is the only coefficient of association that can safely be referred to as simply the "correlation coefficient". It is common enough so that if no other information is provided, it is reasonable to assume that is what is meant.
Correlation coefficients are numbers that give information about the strength of relationship between two variables, such as two different test scores from a sample of participants. The coefficient ranges from -1 through +1. Coefficients between 0 and +1 indicate a positive relationship between the two scores, such as high scores on one test tending to come from people with high scores on the second. The other possible relationship, which is every bit as useful, is a negative correlation between -1 and 0. A negative correlation possesses no less predictive power between the two scores. The difference is that high scores on one measure are associated with low scores on the other.
An example of the kinds of measures that might correlate negatively is absences and grades. People with higher absences will be expected to have lower grades. When a correlation is said to be significant, it can be shown that the correlation is significantly different form zero in the population. A correlation of zero means no relationship between variables. A correlation other than zero means the variables are related. As the coefficient gets further from zero (toward +1 or -1), the relationship becomes stronger.Interpreting Correlation: Magnitude and Sign
Interpreting a Pearson's correlation coefficient (rXY) requires an understanding of two concepts:
· Magnitude.
· Sign (+/-).
The magnitude refers to the strength of the linear relationship between Variable X and Variable
The rXY ranges in values from -1.00 to +1.00. To determine magnitude, ignore the sign of the correlation, and the absolute value of rXY indicates the extent to which Variable X and Variable Y are linearly related. For correlations close to 0, there is no linear relationship. As the correlation approaches either -1.00 or +1.00, the magnitude of the correlation increases. Therefore, for example, the magnitude of r = -.65 is greater than the magnitude of r = +.25 (|.65| > |.25|).
In contrast to magnitude, the sign of a non-zero correlation is either negative or positive.
These labels are not interpreted .
HOW IS IT USEFUL IN FIELD OF FORENSIC SCIENCE AND IN THIS I HAVE SHOWN THE TYPES OF CORRELATION, SIGNIFICANCE , METHODS AND KARL PEARSON'S METHOD OF CORRELATION
This document discusses correlation and regression analysis techniques used in physical geography to examine relationships between variables. Correlation determines the degree of relationship between two variables and is represented by the correlation coefficient r, which ranges from -1 to 1. Regression identifies relationships between a dependent variable and one or more independent variables by calculating a best-fit line that minimizes residuals. The document provides examples of calculating the correlation coefficient r and estimating the regression equation between variables.
Linear regression analysis allows researchers to predict scores on a dependent or criterion variable (Y) based on knowledge of an independent or predictor variable (X). Simple linear regression involves using one predictor variable to predict scores on the dependent variable. Multiple regression expands this to use multiple predictor variables. Key aspects of regression analysis covered in the document include the correlation between variables, using the least squares method to determine the best fitting regression line, computing predicted Y scores, explaining and unexplained variance, and the importance of multiple regression in understanding how well predictor variables predict the criterion variable.
Regression analysis is used to establish relationships between variables and make predictions. It can be used to estimate dependent variables from independent variables, extend analysis to multiple variables, and show the nature of relationships. The key objectives are establishing if relationships exist and making forecasts. Regression requires interval scale data and establishes parameters and an error term in the regression equation. The least squares method chooses parameters that minimize errors between observed and estimated dependent variable values. Goodness of fit is measured by R-squared and F-tests and t-tests determine statistical significance.
This document discusses correlation and defines it as the statistical relationship between two variables, where a change in one variable results in a corresponding change in the other. It describes different types of correlation including positive, negative, simple, partial and multiple. Methods for studying correlation are also outlined, including scatter diagrams and Karl Pearson's coefficient of correlation (represented by r), which quantifies the strength and direction of the linear relationship between two variables from -1 to 1. The coefficient of determination (r2) is also introduced, which expresses the proportion of variance in one variable that is predictable from the other.
The document discusses correlation analysis and correlation coefficients. It provides the following key points:
- Correlation coefficients measure the strength and direction of association between two quantitative variables. The coefficient r ranges from -1 to 1.
- An analysis of height and muscle strength data from 41 male alcoholics yielded a positive correlation coefficient of 0.42, indicating taller men tended to be stronger.
- Similarly, an analysis of the same muscle strength data against age found a negative correlation coefficient of -0.42, suggesting older men tended to be weaker.
This document discusses correlation analysis and its various types. Correlation is a measure of the relationship between two or more variables. There are three main types of correlation based on the degree, number of variables, and linearity. Correlation can be positive, negative, simple, partial, multiple, linear, or non-linear. Correlation is important for understanding relationships between variables, making predictions, and interpreting data. However, correlation does not necessarily imply causation.
This presentation covered the following topics:
1. Definition of Correlation and Regression
2. Meaning of Correlation and Regression
3. Types of Correlation and Regression
4. Karl Pearson's methods of correlation
5. Bivariate Grouped data method
6. Spearman's Rank correlation Method
7. Scattered diagram method
8. Interpretation of correlation coefficient
9. Lines of Regression
10. regression Equations
11. Difference between correlation and regression
12. Related examples
This document defines and explains different types of correlation. It begins by defining correlation as a statistical tool to measure the relationship between two variables. There are three main types of correlation discussed: positive correlation where both variables move in the same direction, negative correlation where the variables move in opposite directions, and zero correlation where a change in one variable does not affect the other. The document also discusses linear and non-linear correlation, as well as simple, partial, and multiple correlation. Different methods for measuring correlation are presented, including graphical methods like scatter diagrams and algebraic methods like Pearson's correlation coefficient.
The document discusses different types and methods of measuring correlation between two variables. It describes Karl Pearson's coefficient of correlation (r) which measures the strength and direction of a linear relationship between two variables on a scale of -1 to 1. It also discusses Spearman's rank correlation coefficient (R) which is used when variables can only be ranked rather than measured quantitatively. The key methods covered are scatter diagrams, which graphically depict relationships, and calculating correlation coefficients based on deviations from the mean.
Multivariate Analysis Degree of association between two variable- Test of Ho...NiezelPertimos
The document discusses multivariate analysis and correlation. It defines correlation as a measure of the degree of association between two variables. A correlation coefficient between -1 and 1 indicates the strength and direction of the linear relationship, with values closer to 1 or -1 being stronger. Positive correlation means the variables move in the same direction, while negative correlation means they move in opposite directions. The document provides examples and methods for calculating and interpreting correlation coefficients, including using scatter plots and the Pearson product-moment formula. Excel functions for finding correlation across multiple data sets are also described.
This document discusses linear regression analysis. Regression analysis measures the relationship between two quantitative variables and can be used to make causal inferences. A regression model shows how dependent and independent variables are related. A bivariate model has one independent variable, while a multivariate model has two or more. Scatterplots graph the relationship between variables. The regression equation specifies the linear relationship between a dependent variable Y and independent variable X. The goal of regression is to find the line that best fits the data by minimizing distances between data points and the line. R-squared indicates how well the regression model predicts observed values, with higher R-squared indicating more of the variance is explained.
❻❸❼⓿❽❻❷⓿⓿❼ SATTA MATKA DPBOSS KALYAN MATKA RESULTS KALYAN CHART KALYAN MATKA MATKA RESULT KALYAN MATKA TIPS SATTA MATKA MATKA COM MATKA PANA JODI TODAY
❻❸❼⓿❽❻❷⓿⓿❼ SATTA MATKA DPBOSS KALYAN MATKA RESULTS KALYAN CHART KALYAN MATKA MATKA RESULT KALYAN MATKA TIPS SATTA MATKA MATKA COM MATKA PANA JODI TODAY
2. INTRODUCTION TO LINEAR EQUATIONS AND
REGRESSION
In the previous chapter, we introduced the Pearson correlation as a technique for describing and measuring the
linear relationship between two variables. Figure 16.1 presents hypothetical data showing the relationship
between SAT scores and college grade point average (GPA). Note that the figure shows a good, but not perfect,
positive relationship. Also note that we have drawn a line through the middle of the data points. This line serves
several purposes:
1. The line makes the relationship between SAT scores and GPA easier to see.
2. The line identifies the center, or central tendency, of the relationship, just as the mean describes central
tendency for a set of scores. Thus, the line provides a simplified description of the relationship. For example, if
the data points were removed, the straight line would still give a general picture of the relationship between SAT
scores and GPA.
3. Finally, the line can be used for prediction. The line establishes a precise, one-to one relationship between
each X value (SAT score) and a corresponding Y value (GPA).
4. Goal
❑Our goal in this section is to develop a procedure that identifies and defines the straight line
that provides the best fit for any specific set of data. This straight line does not have to be drawn
on a graph; it can be presented in a simple equation. Thus, our goal is to find the equation for
the line that best describes the relationship for a set of X and Y data.
❑In general, a linear relationship between two variables X and Y can be expressed by the equation
❑Y = bX + a
❑where a and b are fixed constants. In the general linear equation, the value of b is called the
slope. The slope determines how much the Y variable changes when X is increased by 1 point.
❑Because a straight line can be extremely useful for describing a relationship between two
variables, a statistical technique has been developed that provides a standardized method for
determining the best-fitting straight line for any set of data. The statistical procedure is
regression, and the resulting straight line is called the regression line.
5. The goal for regression is to find the best-fitting straight line for a set of data. To accomplish this
goal, however, it is first necessary to define precisely what is meant by “best fit.”
For any particular set of data, it is possible to draw lots of different straight lines that all appear
to pass through the center of the data points.
Each of these lines can be defined by a linear equation of the form Y = bX + a
where b and a are constants that determine the slope and Y-intercept of the line, respectively.
Each individual line has its own unique values for b and a.
The problem is to find the specific line that provides the best fit to the actual data points. That
we try to find with regression equation.
6. Correlation and Regression
the sign of the correlation (+ or -) is the same as the sign of the slope of the regression line.
Specifically, if the correlation is positive, then the slope is also positive and the regression line
slopes up to the right.
On the other hand, if the correlation is negative, then the slope is negative and the line slopes
down to the right.
A correlation of zero means that the slope is also zero and the regression equation produces a
horizontal line that passes through the data.
7. Interpretation
The predicted value is not perfect (unless r= +1.00 or -1.00).
If you examine Figure 16.4, it should be clear that the data
points do not fit perfectly on the line. In general, there is
some error between the predicted Y values (on the line) and
the actual data. Although the amount of error varies from
point to point, on average the errors are directly related to
the magnitude of the correlation. With a correlation near 1.00
(or -1.00), the data points generally are clustered close to the
line and the error is small.
As the correlation gets nearer to zero, the points move away
from the line and the magnitude of the error increases.
8. Figure 16.5 shows two different sets of data that have exactly the same regression equation. In one case, there
is a perfect correlation (r=+1) between X and Y, so the linear equation fits the data perfectly. For the second set
of data, the predicted Y values on the line only approximate the real data points.
9. It is possible to determine a regression equation for any set of data by simply using the formulas already
presented. The linear equation you obtain is then used to generate predicted Y values for any known
value of X.
However, it should be clear that the accuracy of this prediction depends on how well the points on the
line correspond to the actual data points—that is, the amount of error between the predicted values,
Yˆ, and the actual scores, Y values.
A regression equation, by itself, allows you to make predictions, but it does not provide any information
about the accuracy of the predictions. To measure the precision of the regression, it is customary to
compute a standard error of estimate.
Conceptually, the standard error of estimate is very much like a standard deviation: Both provide a
measure of standard distance. Also, the calculation of the standard error of estimate is very similar to
the calculation of standard deviation.
10. Each deviation measures the distance between the actual Y value (from the data) and the
predicted Y value (from the regression line). This sum of squares is commonly called SS
residual because it is based on the remaining distance between the actual Y scores and the
predicted values.
The obtained SS value is then divided by its degrees of freedom to obtain a measure of
variance.
The degrees of freedom for the standard error of estimate are df = n - 2. The reason for
having n-2 degrees of freedom, rather than the customary n - 1, is that we now are trying to
find the equation for the regression line and you must know the means for both the X and
the Y scores. Specifying these two means places two restrictions on the variability of the
data, with the result that the scores have only n - 2 degrees of freedom.
11. The standard error of estimate are closely related to the value of the correlation. With a large correlation
(near +1.00 or -1.00), the data points are close to the regression line, and the standard error of estimate is
small.
As a correlation gets smaller (near zero), the data points move away from the regression line, and the
standard error of estimate gets larger. Because it is possible to have the same regression equation for
several different sets of data, it is also important to consider r2 and the standard error of estimate.
The regression equation simply describes the best-fitting line and is used for making predictions.
However, r2 and the standard error of estimate indicate how accurate these predictions are.
Earlier (p. 524), we observed that squaring the correlation provides a measure of the accuracy of
prediction. The squared correlation, r2, is called the coefficient of determination because it determines
what proportion of the variability in Y is predicted by the relationship with X. Because r2 measures the
predicted portion of the variability in the Y scores, we can use the expression (1 # r2) to measure the
unpredicted portion. Thus,
predicted variability = SS regression = r2SSY
unpredicted variability = SS residual = (1 - r2)SSY
For example, if r = 0.80, then the predicted variability is r2 = 0.64 (or 64%) of the total variability for the Y
scores and the remaining 36% (1 - r2) is the unpredicted variability. Note that when r = 1.00, the
prediction is perfect and there are no residuals.
12. analysis of regression
The process of testing the significance of a regression equation is called analysis of regression
and is very similar to the analysis of variance (ANOVA.
As with ANOVA, the regression analysis uses an F-ratio to determine whether the variance
predicted by the regression equation is significantly greater than would be expected if there
were no relationship between X and Y.
The F-ratio is a ratio of two variances, or mean square (MS) values, and each variance is
obtained by dividing an SS value by its corresponding degrees of freedom.
The numerator of the F-ratio is MS regression, which is the variance in the Y scores that is
predicted by the regression equation. This variance measures the systematic changes in Y that
occur when the value of X increases or decreases.
The denominator is MS residual, which is the unpredicted variance in the Y scores. This variance
measures the changes in Y that are independent of changes in X.
15. Evaluating The Contribution Of Each
Predictor Variable
In addition to evaluating the overall significance of the multiple-regression equation,
researchers are often interested in the relative contribution of each of the two predictor
variables.
Is one of the predictors responsible for more of the prediction than the other?
However, in the standardized form of the regression equation, the relative size of the beta
values is an indication of the relative contribution of the two variables.
The larger beta value for the X1 predictor indicates that X1 predicts more of the variance than
does X2. The signs of the beta values are also meaningful.
If both betas are positive this indicates that the both X1 and X2 are positively related to Y.
16. Multiple Regression and Partial
Correlations
partial correlation as a technique for measuring the relationship between two variables while
eliminating the influence of a third variable. At that time, we noted that partial correlations serve two
general purposes:
1. A partial correlation can demonstrate that an apparent relationship between two variables is
actually caused by a third variable. Thus, there is no direct relationship between the original two
variables.
2. Partial correlation can demonstrate that there is a relationship between two variables even after a
third variable is controlled. Thus, there really is a relationship between the original two variables that
is not being caused by a third variable.
Multiple regression provides an alternative procedure for accomplishing both of these goals.
Specifically, the regression analysis evaluates the contribution of each predictor variable after the
influence of the other predictor has been considered. Thus, you can determine whether each
predictor variable contributes to the relationship by itself or simply duplicates the contribution
already made by another variable.