# NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )

Size: px
Start display at page:

Download "NCSS Statistical Software Principal Components Regression. In ordinary least squares, the regression coefficients are estimated using the formula ( )"

Transcription

1 Chapter 340 Principal Components Regression Introduction is a technique for analyzing multiple regression data that suffer from multicollinearity. When multicollinearity occurs, least squares estimates are unbiased, but their variances are large so they may be far from the true value. By adding a degree of bias to the regression estimates, principal components regression reduces the standard errors. It is hoped that the net effect will be to give more reliable estimates. Another biased regression technique, ridge regression, is also available in NCSS. Ridge regression is the more popular of the two methods. Multicollinearity Multicollinearity is discussed both in the Multiple Regression chapter and in the Ridge Regression chapter, so we will not repeat the discussion here. However, it is important to understand the impact of multicollinearity so that you can decide if some evasive action (like pc regression) would be beneficial. Models Following the usual notation, suppose our regression equation may be written in matrix form as Y = XB + e where Y is the dependent variable, X represents the independent variables, B is the regression coefficients to be estimated, and e represents the errors or residuals. Standardization The first step is to standardize the variables (both dependent and independent) by subtracting their means and dividing by their standard deviations. This causes a challenge in notation, since we must somehow indicate whether the variables in a particular formula are standardized or not. To keep the presentation simple, we will make the following general statement and then forget about standardization and its confusing notation. As far as standardization is concerned, all calculations are based on standardized variables. When the final regression coefficients are displayed, they are adjusted back to their original scale

2 PC Regression Basics In ordinary least squares, the regression coefficients are estimated using the formula ( ) 1 B = X'X X'Y Note that since the variables are standardized, X X = R, where R is the correlation matrix of independent variables. To perform principal components (PC) regression, we transform the independent variables to their principal components. Mathematically, we write X' X = PDP' = Z'Z where D is a diagonal matrix of the eigenvalues of X X, P is the eigenvector matrix of X X, and Z is a data matrix (similar in structure to X) made up of the principal components. P is orthogonal so that P P = I. We have created new variables Z as weighted averages of the original variables X. This is nothing new to us since we are used to using transformations such as the logarithm and the square root on our data values prior to performing the regression calculations. Since these new variables are principal components, their correlations with each other are all zero. If we begin with variables X1, X2, and X3, we will end up with Z1, Z2, and X3. Severe multicollinearity will be detected as very small eigenvalues. To rid the data of the multicollinearity, we omit the components (the z s) associated with small eigenvalues. Usually, only one or two relatively small eigenvalues will be obtained. For example, if only one small eigenvalue were detected on a problem with three independent variables, we would omit Z3 (the third principal component). When we regress Y on Z1 and Z2, multicollinearity is no longer a problem. We can then transform our results back to the X scale to obtain estimates of B. These estimates will be biased, but we hope that the size of this bias is more than compensated for by the decrease in variance. That is, we hope that the mean squared error of these estimates is less than that for least squares. Mathematically, the estimation formula becomes ( ) 1 1 A = Z'Z Z'Y = D Z'Y because of the special nature of principal components. Notice that this is ordinary least squares regression applied to a different set of independent variables. The two sets of regression coefficients, A and B, are related using the formulas and A = P'B B = PA Omitting a principal component may be accomplished by setting the corresponding element of A equal to zero. Hence, the principal components regression may be outlined as follows: 1. Complete a principal components analysis of the X matrix and save the principal components in Z. 2. Fit the regression of Y on Z obtaining least squares estimates of A. 3. Set the last element of A equal to zero. 4. Transform back to the original coefficients using B = PA. Alternative Interpretation of PC Regression It can be shown that omitting a principal component amounts to setting a linear constraint on the regression coefficients. That is, in the case of three independent variables, we add the constraint 340-2

3 p13b1 + p23b2 + p33b3 = 0 Note that this is a constraint on the coefficients, not a constraint on the dependent variable. Essentially, we have avoided the multicollinearity problem by avoiding the region of the solution space in which it occurs. How Many PC s Should Be Omitted Unlike the selection of k in ridge regression, the selection of the number of PC s to omit is relatively straight forward. We omit the PC s corresponding to small eigenvalues. Since the size of the typical eigenvalue of a correlation matrix is one, we omit those that are much smaller than one. Usually, the choice will be obvious. Assumptions The assumptions are the same as those used in regular multiple regression: linearity, constant variance (no outliers), and independence. Since PC regression does not provide confidence limits, normality need not be assumed. Data Structure The data are entered as three or more variables. One variable represents the dependent variable. The other variables represent the independent variables. An example of data appropriate for this procedure is shown below. These data were concocted to have a high degree of multicollinearity as follows. We put a sequence of numbers in X1. Next, we put another series of numbers in X3 that were selected to be unrelated to X1. We created X2 by adding X1 and X3. We made a few changes in X2 so that there was not perfect correlation. Finally, we added all three variables and some random error to form Y. The data are contained in the RidgeReg dataset. We suggest that you open this database now so that you can follow along with the example. RidgeReg dataset (subset) X1 X2 X3 Y

4 Missing Values Rows with missing values in the variables being analyzed are ignored. If data are present on a row for all but the dependent variable, a predicted value is generated for that row. Procedure Options This section describes the options available in this procedure. Variables Tab This panel specifies the variables used in the analysis. Dependent Variables Y: Dependent Variable(s) Specifies a dependent (Y) variable. If more than one variable is specified, a separate analysis is run for each. Weight Variable Weight Variable Specifies a variable containing observation (row) weights for generating weighted-regression analysis. Rows which have zero or negative weights are dropped. Independent Variables X s: Independent Variables Specifies the variable(s) to be used as independent (X) variables. Estimation Options PC s Omitted This is the number of principal components that are omitted during the estimation procedure. PC regression is most useful in those cases where this value is set to one or two. If more than two eigenvalues are small, you probably should take some other evasive action such as completely removing the offending variable(s) from consideration. Reports Tab The following options control which reports and plots are displayed. Select Reports Descriptive Statistics... Predicted Values & Residuals These options specify which reports are displayed

5 Report Options Precision Specifies the precision of numbers in the report. Single precision will display seven-place accuracy, while the double precision will display thirteen-place accuracy. Variable Names This option lets you select whether to display variable names, variable labels, or both. Report Options Decimal Places Beta... VIF Decimals Each of these options specifies the number of decimal places to display for that particular item. Plots Tab The options on this panel control the inclusion and the appearance of the various plots. Select Plots Beta Trace... Residuals vs X's These options specify which plots are displayed. Click the plot format button to change the plot settings. Storage Tab Predicted values and residuals may be calculated for each row and stored on the current dataset for further analysis. The selected statistics are automatically stored to the current dataset. Note that existing data are replaced. Also, if you specify more than one dependent variable, you should specify a corresponding number of storage columns here. Following is a description of the statistics that can be stored. Data Storage Columns Predicted Values The predicted (Yhat) values. Residuals The residuals (Y-Yhat)

6 Example 1 This section presents an example of how to run a principal components regression analysis of the data presented above. The data are in the RidgeReg dataset. In this example, we will run a regression of Y on X1 - X3. You may follow along here by making the appropriate entries or load the completed template Example 1 by clicking on Open Example Template from the File menu of the window. 1 Open the RidgeReg dataset. From the File menu of the NCSS Data window, select Open Example Data. Click on the file RidgeReg.NCSS. Click Open. 2 Open the window. On the menus, select Analysis, then Regression, then. The Principal Components Regression procedure will be displayed. On the menus, select File, then New Template. This will fill the procedure with the default template. 3 Specify the variables. On the window, select the Variables tab. Double-click in the Y: Dependent Variable(s) text box. This will bring up the variable selection window. Select Y from the list of variables and then click Ok. Y will appear in the Y: Dependent Variable(s) box. Double-click in the X s: Independent Variables text box. This will bring up the variable selection window. Select X1 - X3 from the list of variables and then click Ok. X1-X3 will appear in the X s: Independent Variables. 4 Specify the reports. Select the Reports and Plots tabs. Check all reports and plots. All are selected here for documentation purposes. Some of the aspects of the plot axes may be modified for improved viewing. 5 Run the procedure. From the Run menu, select Run Procedure. Alternatively, just click the green Run button. Descriptive Statistics Section Descriptive Statistics Section Standard Variable Count Mean Deviation Minimum Maximum X X X Y For each variable, the descriptive statistics of the nonmissing values are computed. This report is particularly useful for checking that the correct variables were selected

7 Correlation Matrix Section Correlation Matrix Section X1 X2 X3 Y X X X Y Pearson correlations are given for all variables. Outliers, nonnormality, nonconstant variance, and nonlinearities can all impact these correlations. Note that these correlations may differ from pair-wise correlations generated by the correlation matrix program because of the different ways the two programs treat rows with missing values. The method used here is row-wise deletion. These correlation coefficients show which independent variables are highly correlated with the dependent variable and with each other. Independent variables that are highly correlated with one another may cause multicollinearity problems. Least Squares Multicollinearity Section Least Squares Multicollinearity Section Independent Variance R-Squared Variable Inflation Vs Other X s Tolerance X X X Since some VIF s are greater than 10, multicollinearity is a problem. This report provides information useful in assessing the amount of multicollinearity in your data. Variance Inflation The variance inflation factor (VIF) is a measure of multicollinearity. It is the reciprocal of 1-R x2, where R x 2 is the R 2 obtained when this variable is regressed on the remaining independent variables. A VIF of 10 or more for large data sets indicates a multicollinearity problem since the R x 2 with the remaining X s is 90 percent. For small data sets, even VIF s of 5 or more can signify multicollinearity. VIF j = R R-Squared vs Other X s R x 2 is the R-squared obtained when this variable is regressed on the remaining independent variables. A high R x 2 indicates a lot of overlap in explaining the variation among the remaining independent variables. Tolerance Tolerance is just 1- R x2, the denominator of the variance inflation factor. 2 j 340-7

8 Eigenvalues of Correlations Eigenvalues of Correlations Incremental Cumulative Condition No. Eigenvalue Percent Percent Number Some Condition Numbers greater than Multicollinearity is a SEVERE problem. This section gives an eigenvalue analysis of the independent variables after they have been centered and scaled. Notice that in this example, the third eigenvalue is very small. Eigenvalue The eigenvalues of the correlation matrix. The sum of the eigenvalues is equal to the number of independent variables. Eigenvalues near zero indicate a multicollinearity problem in your data. Incremental Percent Incremental percent is the percent this eigenvalue is of the total. In an ideal situation, these percentages would be equal. Percents near zero indicate a multicollinearity problem in your data. Cumulative Percent This is the running total of the Incremental Percent. Condition Number The condition number is the largest eigenvalue divided by each corresponding eigenvalue. Since the eigenvalues are really variances, the condition number is a ratio of variances. Condition numbers greater than 1000 indicate a severe multicollinearity problem while condition numbers between 100 and 1000 indicate a mild multicollinearity problem. Eigenvector of Correlations Eigenvector of Correlations No. Eigenvalue X1 X2 X This report displays the eigenvectors associated with each eigenvalue. The notion behind eigenvalue analysis is that the axes are rotated from those defined by the variables to a new set defined by the variances of the variables. Rotation is accomplished by taking weighted averages of the standardized original variables. The first new variable is constructed to account for the largest amount of variance possible from a single axis. No. The number of the eigenvalue. Eigenvalue The eigenvalues of the correlation matrix. The sum of the eigenvalues is equal to the number of independent variables. Eigenvalues near zero indicate multicollinearity in your data. The eigenvalues represent the spread (variance) in the direction defined by this new axis. Hence, small eigenvalues indicate directions in which there is no spread. Since regression analysis seeks to find trends across values, when there is not a spread, the trends cannot be computed accurately

9 Table-Values The table values give the eigenvectors. The eigenvectors give the weights that are used to create the new axis. By studying the weights, you can gain an understanding of what is happening in the data. In the example above, we can see that the first factor (new variable associated with the first eigenvalue) is constructed by adding X1 and X2. Note that the weights are almost equal. X3 has a small weight, indicating that it does not play a role in this factor. Factor 2 seems to be completely created from X3. X1 and X2 play only a small role in its construction. Factor 3 seems to be the difference between X1 and X2. Again X3 plays only a small role. Hence, the interpretation of these eigenvectors leads to the following statements: 1. Most of the variation in X1, X2, and X3 can be accounted for by considering only two variables: Z = X1+X2 and X3. 2. The third dimension, calculated as X1-X2, is almost negligible and might be ignored. Beta Trace Section This plot shows the standardized regression coefficients (often referred to as the betas) on the vertical axis and the number of principal components (PC s) included along the horizontal axis. Thus, the set on the right is the least squares set. By studying this plot, you can determine what omitting a certain number of PC s has done to the estimated regression coefficients

10 Variance Inflation Factor Plot This is a plot that shows the effect of the omitted PC s on the variance inflation factors. Since the major goal of PC regression is to remove the impact of multicollinearity, it is important to know at what point multicollinearity has been dealt with. This plot shows this. Since the rule-of-thumb is that multicollinearity is not a problem once all VIFs are less than 10, we inspect the graph for this point. In this example, it appears that all VIFs are less than 10 if only two of the three PC s are included. Standardized Regression Coefficients Section Standardized Regression Coefficients Section PC s X1 X2 X This report gives the values that are plotted on the beta trace. Variance Inflation Factor Section Variance Inflation Factor Section PC s X1 X2 X This report gives the values that are plotted on the variance inflation factor plot. Note how easy it is to determine when all three VIFs are less than

11 Components Analysis Section Components Analysis Section PC s R2 Sigma B'B Ave VIF Max VIF This report provides a quick summary of the various statistics that might go into the choice of k. PC s This is the number of principal components included in the regression reported on this line. R2 This is the value of R-squared. Since the least squares solution maximizes R-squared, the largest value of R- squared occurs at bottom of the report (when all PC s are included). Sigma This is the square root of the mean squared error. Least squares minimizes this value, so we want to select the number of PC s that does not stray very much from the least squares value. B B This is the sum of the squared standardized regression coefficients. PC regression assumes that this value is too large and so the method tries to reduce this. We want to find the number of PC s at which this value has stabilized. Ave VIF This is the average of the variance inflation factors. Max VIF This is the maximum variance inflation factor. Since we are looking for the number of PC s which results in all VIFs being less than 10, this value is very helpful. P.C. versus L.S. Comparison Section P.C. vs. Least Squares Comparison Section with 1 Component Omitted Regular Regular Stand'zed Stand'zed Component L.S. Independent Component L.S. Component L.S. Standard Standard Variable Coeff's Coeff's Coeff's Coeff's Error Error Intercept X X E X R-Squared Sigma This report provides a detailed comparison between the PC regression solution and the ordinary least squares solution to the estimation of the regression coefficients. Independent Variable The names of the independent variables are listed here. The intercept is the value of b 0. Regular Component (and L.S.) Coeff s These are the estimated values of the regression coefficients b 0, b 1,..., b p. The first column gives the values for PC regression and the second column gives the values for regular least squares regression

12 The value indicates how much change in Y occurs for a one-unit change in X when the remaining X s are held constant. These coefficients are also called partial-regression coefficients since the effect of the other X s is removed. Stand zed Component (and L.S.) Coeff s These are the estimated values of the standardized regression coefficients. The first column gives the values for PC regression and the second column gives the values for regular least squares regression. Standardized regression coefficients are the coefficients that would be obtained if you standardized each independent and dependent variable. Here standardizing is defined as subtracting the mean and dividing by the standard deviation of a variable. A regression analysis on these standardized variables would yield these standardized coefficients. When there are vastly different units involved for the variables, this is a way of making comparisons between variables. The formula for the standardized regression coefficient is: b = b j, std j where s y and s x j are the standard deviations for the dependent variable and the corresponding j th independent variable. Component (and L.S.) Standard Error These are the estimated standard errors (precision) of the regression coefficients. The first column gives the values for PC regression and the second column gives the values for regular least squares regression. The standard error of the regression coefficient, s b j, is the standard deviation of the estimate. Since one of the objects of PC regression is to reduce this (make the estimates more precise), it is of interest to see how much reduction has taken place. R-Squared R-squared is the coefficient of determination. It represents the percent of variation in the dependent variable explained by the independent variables in the model. The R-squared values of both the PC and regular regressions are shown. Sigma This is the square root of the mean squared error. It provides a measure of the standard deviation of the residuals from the regression model. It represents the percent of variation in the dependent variable explained by the independent variables in the model. The R-squared values of both the PC and regular regressions are shown. s y sx j

13 PC Coefficient Section PC Coefficient Section Principal PC Individual Component Coefficient R-Squared Eigenvalue PC PC PC This report provides the details of the regression based on the principal components (the Z s). Principal Component This is the number of the principal component being reported about on this line. The order here corresponds to the order of the eigenvalues. Thus, the first is associated with the largest eigenvalue and the last is associated with the smallest. PC Coefficient These are the estimated values of the regression coefficients a 1,..., a p. The value indicates how much change in Y occurs for a one-unit change in z when the remaining z s are held constant. Individual R-Squared This is the amount contributed to R-squared by this component. Eigenvalue This is the eigenvalue of this component. PC Regression Coefficient Section Regression Coefficient Section with 1 Component Omitted Stand'zed Independent Regression Standard Regression Variable Coefficient Error Coefficient VIF Intercept X X E X This report provides the details of the PC regression solution. Independent Variable The names of the independent variables are listed here. The intercept is the value of b 0. Regression Coefficient These are the estimated values of the regression coefficients b 0, b 1,..., b p. The value indicates how much change in Y occurs for a one-unit change in x when the remaining X s are held constant. These coefficients are also called partial-regression coefficients since the effect of the other X s is removed. Standard Error These are the estimated standard errors (precision) of the PC regression coefficients. The standard error of the regression coefficient, s b j, is the standard deviation of the estimate. In regular regression, we divide the coefficient by the standard error to obtain a t statistic. However, this is not possible here because of the bias in the estimates. Stand zed Regression Coefficient These are the estimated values of the standardized regression coefficients. Standardized regression coefficients are the coefficients that would be obtained if you standardized each independent and dependent variable. Here

14 standardizing is defined as subtracting the mean and dividing by the standard deviation of a variable. A regression analysis on these standardized variables would yield these standardized coefficients. When there are vastly different units involved for the variables, this is a way of making comparisons between variables. The formula for the standardized regression coefficient is: b = b j, std j where s y and s x j are the standard deviations for the dependent variable and the corresponding j th independent variable. VIF These are the values of the variance inflation factors associated with the variables. When multicollinearity has been conquered, these values will all be less than 10. Details of what VIF is were given earlier. s y sx j Analysis of Variance Section Analysis of Variance Section with 1 Component Omitted Sum of Mean Prob Source DF Squares Square F-Ratio Level Intercept Model Error Total(Adjusted) Mean of Dependent Root Mean Square Error R-Squared Coefficient of Variation E-02 An analysis of variance (ANOVA) table summarizes the information related to the sources of variation in the data. Source This represents the partitions of the variation in y. There are four sources of variation listed: intercept, model, error, and total (adjusted for the mean). DF The degrees of freedom are the number of dimensions associated with this term. Note that each observation can be interpreted as a dimension in n-dimensional space. The degrees of freedom for the intercept, model, error, and adjusted total are 1, p, n-p-1, and n-1, respectively. Sum of Squares These are the sums of squares associated with the corresponding sources of variation. Note that these values are in terms of the dependent variable, y. The formulas for each are: Mean Square The mean square is the sum of squares divided by the degrees of freedom. This mean square is an estimated variance. For example, the mean square error is the estimated variance of the residuals (the residuals are sometimes called the errors). F-Ratio This is the F statistic for testing the null hypothesis that all β j = 0. This F-statistic has p degrees of freedom for the numerator variance and n-p-1 degrees of freedom for the denominator variance

15 Since PC regression produces biased estimates, this F-Ratio is not a valid test. It serves as an index, but it would not stand up under close scrutiny. Prob Level This is the p-value for the above F test. The p-value is the probability that the test statistic will take on a value at least as extreme as the observed value, assuming that the null hypothesis is true. If the p-value is less than α, say 0.05, the null hypothesis is rejected. If the p-value is greater than α, then the null hypothesis is accepted. Root Mean Square Error This is the square root of the mean square error. It is an estimate of σ, the standard deviation of the e i s. Mean of Dependent Variable This is the arithmetic mean of the dependent variable. R-Squared This is the coefficient of determination. It is defined in full in the Multiple Regression chapter. Coefficient of Variation The coefficient of variation is a relative measure of dispersion, computed by dividing root mean square error by the mean of the dependent variable. By itself, it has little value, but it can be useful in comparative studies. CV = MSE y Predicted Values and Residuals Section Predicted Values and Residuals Section with 1 Component Omitted Row Actual Predicted Residual This section reports the predicted values and the sample residuals, or e i s. When you want to generate predicted values for individuals not in your sample, add their values to the bottom of your database, leaving the dependent variable blank. Their predicted values will be shown on this report. Actual This is the actual value of Y for the i th row. Predicted The predicted value of Y for the i th row. It is predicted using the levels of the X s for this row

16 Residual This is the estimated value of e i. This is equal to the Actual minus the Predicted. Histogram The purpose of the histogram and density trace of the residuals is to display the distribution of the residuals. The odd shape of this histogram occurs because of the way in which these particular data were manufactured. Probability Plot of Residuals

17 Residual vs Predicted Plot This plot should always be examined. The preferred pattern to look for is a point cloud or a horizontal band. A wedge or bowtie pattern is an indicator of nonconstant variance, a violation of a critical regression assumption. A sloping or curved band signifies inadequate specification of the model. A sloping band with increasing or decreasing variability suggests nonconstant variance and inadequate specification of the model. Residual vs Predictor(s) Plot This is a scatter plot of the residuals versus each independent variable. Again, the preferred pattern is a rectangular shape or point cloud. Any other nonrandom pattern may require a redefining of the regression model

### Stepwise Regression. Chapter 311. Introduction. Variable Selection Procedures. Forward (Step-Up) Selection

Chapter 311 Introduction Often, theory and experience give only general direction as to which of a pool of candidate variables (including transformed variables) should be included in the regression model.

### Factor Analysis. Chapter 420. Introduction

Chapter 420 Introduction (FA) is an exploratory technique applied to a set of observed variables that seeks to find underlying factors (subsets of variables) from which the observed variables were generated.

### Gamma Distribution Fitting

Chapter 552 Gamma Distribution Fitting Introduction This module fits the gamma probability distributions to a complete or censored set of individual or grouped data values. It outputs various statistics

### Multivariate Analysis of Variance (MANOVA)

Chapter 415 Multivariate Analysis of Variance (MANOVA) Introduction Multivariate analysis of variance (MANOVA) is an extension of common analysis of variance (ANOVA). In ANOVA, differences among various

### Regression Clustering

Chapter 449 Introduction This algorithm provides for clustering in the multiple regression setting in which you have a dependent variable Y and one or more independent variables, the X s. The algorithm

### Lin s Concordance Correlation Coefficient

NSS Statistical Software NSS.com hapter 30 Lin s oncordance orrelation oefficient Introduction This procedure calculates Lin s concordance correlation coefficient ( ) from a set of bivariate data. The

### Simple Regression Theory II 2010 Samuel L. Baker

SIMPLE REGRESSION THEORY II 1 Simple Regression Theory II 2010 Samuel L. Baker Assessing how good the regression equation is likely to be Assignment 1A gets into drawing inferences about how close the

### How To Run Statistical Tests in Excel

How To Run Statistical Tests in Excel Microsoft Excel is your best tool for storing and manipulating data, calculating basic descriptive statistics such as means and standard deviations, and conducting

### 2. Simple Linear Regression

Research methods - II 3 2. Simple Linear Regression Simple linear regression is a technique in parametric statistics that is commonly used for analyzing mean response of a variable Y which changes according

### Dimensionality Reduction: Principal Components Analysis

Dimensionality Reduction: Principal Components Analysis In data mining one often encounters situations where there are a large number of variables in the database. In such situations it is very likely

### Introduction to Regression and Data Analysis

Statlab Workshop Introduction to Regression and Data Analysis with Dan Campbell and Sherlock Campbell October 28, 2008 I. The basics A. Types of variables Your variables may take several forms, and it

### KSTAT MINI-MANUAL. Decision Sciences 434 Kellogg Graduate School of Management

KSTAT MINI-MANUAL Decision Sciences 434 Kellogg Graduate School of Management Kstat is a set of macros added to Excel and it will enable you to do the statistics required for this course very easily. To

### Data Analysis Tools. Tools for Summarizing Data

Data Analysis Tools This section of the notes is meant to introduce you to many of the tools that are provided by Excel under the Tools/Data Analysis menu item. If your computer does not have that tool

### Point Biserial Correlation Tests

Chapter 807 Point Biserial Correlation Tests Introduction The point biserial correlation coefficient (ρ in this chapter) is the product-moment correlation calculated between a continuous random variable

### Regression step-by-step using Microsoft Excel

Step 1: Regression step-by-step using Microsoft Excel Notes prepared by Pamela Peterson Drake, James Madison University Type the data into the spreadsheet The example used throughout this How to is a regression

### Outline. Topic 4 - Analysis of Variance Approach to Regression. Partitioning Sums of Squares. Total Sum of Squares. Partitioning sums of squares

Topic 4 - Analysis of Variance Approach to Regression Outline Partitioning sums of squares Degrees of freedom Expected mean squares General linear test - Fall 2013 R 2 and the coefficient of correlation

### Regression Analysis (Spring, 2000)

Regression Analysis (Spring, 2000) By Wonjae Purposes: a. Explaining the relationship between Y and X variables with a model (Explain a variable Y in terms of Xs) b. Estimating and testing the intensity

### NCSS Statistical Software

Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, two-sample t-tests, the z-test, the

### Doing Multiple Regression with SPSS. In this case, we are interested in the Analyze options so we choose that menu. If gives us a number of choices:

Doing Multiple Regression with SPSS Multiple Regression for Data Already in Data Editor Next we want to specify a multiple regression analysis for these data. The menu bar for SPSS offers several options:

### Scatter Plots with Error Bars

Chapter 165 Scatter Plots with Error Bars Introduction The procedure extends the capability of the basic scatter plot by allowing you to plot the variability in Y and X corresponding to each point. Each

### 5. Linear Regression

5. Linear Regression Outline.................................................................... 2 Simple linear regression 3 Linear model............................................................. 4

### Simple linear regression

Simple linear regression Introduction Simple linear regression is a statistical method for obtaining a formula to predict values of one variable from another where there is a causal relationship between

### Summary of important mathematical operations and formulas (from first tutorial):

EXCEL Intermediate Tutorial Summary of important mathematical operations and formulas (from first tutorial): Operation Key Addition + Subtraction - Multiplication * Division / Exponential ^ To enter a

### Using Excel (Microsoft Office 2007 Version) for Graphical Analysis of Data

Using Excel (Microsoft Office 2007 Version) for Graphical Analysis of Data Introduction In several upcoming labs, a primary goal will be to determine the mathematical relationship between two variable

### Randomized Block Analysis of Variance

Chapter 565 Randomized Block Analysis of Variance Introduction This module analyzes a randomized block analysis of variance with up to two treatment factors and their interaction. It provides tables of

### SPSS Guide: Regression Analysis

SPSS Guide: Regression Analysis I put this together to give you a step-by-step guide for replicating what we did in the computer lab. It should help you run the tests we covered. The best way to get familiar

### Chapter 7: Simple linear regression Learning Objectives

Chapter 7: Simple linear regression Learning Objectives Reading: Section 7.1 of OpenIntro Statistics Video: Correlation vs. causation, YouTube (2:19) Video: Intro to Linear Regression, YouTube (5:18) -

### Pearson's Correlation Tests

Chapter 800 Pearson's Correlation Tests Introduction The correlation coefficient, ρ (rho), is a popular statistic for describing the strength of the relationship between two variables. The correlation

### Exercise 1.12 (Pg. 22-23)

Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.

### Regression Analysis: A Complete Example

Regression Analysis: A Complete Example This section works out an example that includes all the topics we have discussed so far in this chapter. A complete example of regression analysis. PhotoDisc, Inc./Getty

### EXCEL Tutorial: How to use EXCEL for Graphs and Calculations.

EXCEL Tutorial: How to use EXCEL for Graphs and Calculations. Excel is powerful tool and can make your life easier if you are proficient in using it. You will need to use Excel to complete most of your

### Case Study in Data Analysis Does a drug prevent cardiomegaly in heart failure?

Case Study in Data Analysis Does a drug prevent cardiomegaly in heart failure? Harvey Motulsky hmotulsky@graphpad.com This is the first case in what I expect will be a series of case studies. While I mention

### Chapter 13 Introduction to Linear Regression and Correlation Analysis

Chapter 3 Student Lecture Notes 3- Chapter 3 Introduction to Linear Regression and Correlation Analsis Fall 2006 Fundamentals of Business Statistics Chapter Goals To understand the methods for displaing

### Multiple Linear Regression

Multiple Linear Regression A regression with two or more explanatory variables is called a multiple regression. Rather than modeling the mean response as a straight line, as in simple regression, it is

### CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression

Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the

### Simple Linear Regression Inference

Simple Linear Regression Inference 1 Inference requirements The Normality assumption of the stochastic term e is needed for inference even if it is not a OLS requirement. Therefore we have: Interpretation

### NCSS Statistical Software

Chapter 06 Introduction This procedure provides several reports for the comparison of two distributions, including confidence intervals for the difference in means, two-sample t-tests, the z-test, the

### Statistical Models in R

Statistical Models in R Some Examples Steven Buechler Department of Mathematics 276B Hurley Hall; 1-6233 Fall, 2007 Outline Statistical Models Linear Models in R Regression Regression analysis is the appropriate

### Chapter 23. Inferences for Regression

Chapter 23. Inferences for Regression Topics covered in this chapter: Simple Linear Regression Simple Linear Regression Example 23.1: Crying and IQ The Problem: Infants who cry easily may be more easily

: Table of Contents... 1 Overview of Model... 1 Dispersion... 2 Parameterization... 3 Sigma-Restricted Model... 3 Overparameterized Model... 4 Reference Coding... 4 Model Summary (Summary Tab)... 5 Summary

### Review Jeopardy. Blue vs. Orange. Review Jeopardy

Review Jeopardy Blue vs. Orange Review Jeopardy Jeopardy Round Lectures 0-3 Jeopardy Round \$200 How could I measure how far apart (i.e. how different) two observations, y 1 and y 2, are from each other?

### International Statistical Institute, 56th Session, 2007: Phil Everson

Teaching Regression using American Football Scores Everson, Phil Swarthmore College Department of Mathematics and Statistics 5 College Avenue Swarthmore, PA198, USA E-mail: peverso1@swarthmore.edu 1. Introduction

### Binary Diagnostic Tests Two Independent Samples

Chapter 537 Binary Diagnostic Tests Two Independent Samples Introduction An important task in diagnostic medicine is to measure the accuracy of two diagnostic tests. This can be done by comparing summary

### This chapter will demonstrate how to perform multiple linear regression with IBM SPSS

CHAPTER 7B Multiple Regression: Statistical Methods Using IBM SPSS This chapter will demonstrate how to perform multiple linear regression with IBM SPSS first using the standard method and then using the

### ESTIMATING THE DISTRIBUTION OF DEMAND USING BOUNDED SALES DATA

ESTIMATING THE DISTRIBUTION OF DEMAND USING BOUNDED SALES DATA Michael R. Middleton, McLaren School of Business, University of San Francisco 0 Fulton Street, San Francisco, CA -00 -- middleton@usfca.edu

### Please follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software

STATA Tutorial Professor Erdinç Please follow the directions once you locate the Stata software in your computer. Room 114 (Business Lab) has computers with Stata software 1.Wald Test Wald Test is used

### An analysis appropriate for a quantitative outcome and a single quantitative explanatory. 9.1 The model behind linear regression

Chapter 9 Simple Linear Regression An analysis appropriate for a quantitative outcome and a single quantitative explanatory variable. 9.1 The model behind linear regression When we are examining the relationship

### Module 3: Correlation and Covariance

Using Statistical Data to Make Decisions Module 3: Correlation and Covariance Tom Ilvento Dr. Mugdim Pašiƒ University of Delaware Sarajevo Graduate School of Business O ften our interest in data analysis

### 1. What is the critical value for this 95% confidence interval? CV = z.025 = invnorm(0.025) = 1.96

1 Final Review 2 Review 2.1 CI 1-propZint Scenario 1 A TV manufacturer claims in its warranty brochure that in the past not more than 10 percent of its TV sets needed any repair during the first two years

### Below is a very brief tutorial on the basic capabilities of Excel. Refer to the Excel help files for more information.

Excel Tutorial Below is a very brief tutorial on the basic capabilities of Excel. Refer to the Excel help files for more information. Working with Data Entering and Formatting Data Before entering data

### August 2012 EXAMINATIONS Solution Part I

August 01 EXAMINATIONS Solution Part I (1) In a random sample of 600 eligible voters, the probability that less than 38% will be in favour of this policy is closest to (B) () In a large random sample,

### Chapter 7. One-way ANOVA

Chapter 7 One-way ANOVA One-way ANOVA examines equality of population means for a quantitative outcome and a single categorical explanatory variable with any number of levels. The t-test of Chapter 6 looks

### Simple Linear Regression, Scatterplots, and Bivariate Correlation

1 Simple Linear Regression, Scatterplots, and Bivariate Correlation This section covers procedures for testing the association between two continuous variables using the SPSS Regression and Correlate analyses.

### Getting Correct Results from PROC REG

Getting Correct Results from PROC REG Nathaniel Derby, Statis Pro Data Analytics, Seattle, WA ABSTRACT PROC REG, SAS s implementation of linear regression, is often used to fit a line without checking

### Bill Burton Albert Einstein College of Medicine william.burton@einstein.yu.edu April 28, 2014 EERS: Managing the Tension Between Rigor and Resources 1

Bill Burton Albert Einstein College of Medicine william.burton@einstein.yu.edu April 28, 2014 EERS: Managing the Tension Between Rigor and Resources 1 Calculate counts, means, and standard deviations Produce

### 17. SIMPLE LINEAR REGRESSION II

17. SIMPLE LINEAR REGRESSION II The Model In linear regression analysis, we assume that the relationship between X and Y is linear. This does not mean, however, that Y can be perfectly predicted from X.

### Assignment objectives:

Assignment objectives: Regression Pivot table Exercise #1- Simple Linear Regression Often the relationship between two variables, Y and X, can be adequately represented by a simple linear equation of the

### Notes on Applied Linear Regression

Notes on Applied Linear Regression Jamie DeCoster Department of Social Psychology Free University Amsterdam Van der Boechorststraat 1 1081 BT Amsterdam The Netherlands phone: +31 (0)20 444-8935 email:

### Part 2: Analysis of Relationship Between Two Variables

Part 2: Analysis of Relationship Between Two Variables Linear Regression Linear correlation Significance Tests Multiple regression Linear Regression Y = a X + b Dependent Variable Independent Variable

### Multiple Regression in SPSS This example shows you how to perform multiple regression. The basic command is regression : linear.

Multiple Regression in SPSS This example shows you how to perform multiple regression. The basic command is regression : linear. In the main dialog box, input the dependent variable and several predictors.

### Simple Linear Regression

STAT 101 Dr. Kari Lock Morgan Simple Linear Regression SECTIONS 9.3 Confidence and prediction intervals (9.3) Conditions for inference (9.1) Want More Stats??? If you have enjoyed learning how to analyze

### Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression

Unit 31 A Hypothesis Test about Correlation and Slope in a Simple Linear Regression Objectives: To perform a hypothesis test concerning the slope of a least squares line To recognize that testing for a

### Directions for using SPSS

Directions for using SPSS Table of Contents Connecting and Working with Files 1. Accessing SPSS... 2 2. Transferring Files to N:\drive or your computer... 3 3. Importing Data from Another File Format...

### Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written

### Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011

Chicago Booth BUSINESS STATISTICS 41000 Final Exam Fall 2011 Name: Section: I pledge my honor that I have not violated the Honor Code Signature: This exam has 34 pages. You have 3 hours to complete this

### DESCRIPTIVE STATISTICS AND EXPLORATORY DATA ANALYSIS

DESCRIPTIVE STATISTICS AND EXPLORATORY DATA ANALYSIS SEEMA JAGGI Indian Agricultural Statistics Research Institute Library Avenue, New Delhi - 110 012 seema@iasri.res.in 1. Descriptive Statistics Statistics

### Stat 5303 (Oehlert): Tukey One Degree of Freedom 1

Stat 5303 (Oehlert): Tukey One Degree of Freedom 1 > catch

### Multicollinearity Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised January 13, 2015

Multicollinearity Richard Williams, University of Notre Dame, http://www3.nd.edu/~rwilliam/ Last revised January 13, 2015 Stata Example (See appendices for full example).. use http://www.nd.edu/~rwilliam/stats2/statafiles/multicoll.dta,

### X X X a) perfect linear correlation b) no correlation c) positive correlation (r = 1) (r = 0) (0 < r < 1)

CORRELATION AND REGRESSION / 47 CHAPTER EIGHT CORRELATION AND REGRESSION Correlation and regression are statistical methods that are commonly used in the medical literature to compare two or more variables.

### There are six different windows that can be opened when using SPSS. The following will give a description of each of them.

SPSS Basics Tutorial 1: SPSS Windows There are six different windows that can be opened when using SPSS. The following will give a description of each of them. The Data Editor The Data Editor is a spreadsheet

### Recall this chart that showed how most of our course would be organized:

Chapter 4 One-Way ANOVA Recall this chart that showed how most of our course would be organized: Explanatory Variable(s) Response Variable Methods Categorical Categorical Contingency Tables Categorical

### Testing for Lack of Fit

Chapter 6 Testing for Lack of Fit How can we tell if a model fits the data? If the model is correct then ˆσ 2 should be an unbiased estimate of σ 2. If we have a model which is not complex enough to fit

### Comparables Sales Price

Chapter 486 Comparables Sales Price Introduction Appraisers often estimate the market value (current sales price) of a subject property from a group of comparable properties that have recently sold. Since

### An analysis method for a quantitative outcome and two categorical explanatory variables.

Chapter 11 Two-Way ANOVA An analysis method for a quantitative outcome and two categorical explanatory variables. If an experiment has a quantitative outcome and two categorical explanatory variables that

### Data Analysis. Using Excel. Jeffrey L. Rummel. BBA Seminar. Data in Excel. Excel Calculations of Descriptive Statistics. Single Variable Graphs

Using Excel Jeffrey L. Rummel Emory University Goizueta Business School BBA Seminar Jeffrey L. Rummel BBA Seminar 1 / 54 Excel Calculations of Descriptive Statistics Single Variable Graphs Relationships

### Summary of Formulas and Concepts. Descriptive Statistics (Ch. 1-4)

Summary of Formulas and Concepts Descriptive Statistics (Ch. 1-4) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume

### AP Physics 1 and 2 Lab Investigations

AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks

### DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9

DEPARTMENT OF PSYCHOLOGY UNIVERSITY OF LANCASTER MSC IN PSYCHOLOGICAL RESEARCH METHODS ANALYSING AND INTERPRETING DATA 2 PART 1 WEEK 9 Analysis of covariance and multiple regression So far in this course,

### Univariate Regression

Univariate Regression Correlation and Regression The regression line summarizes the linear relationship between 2 variables Correlation coefficient, r, measures strength of relationship: the closer r is

### 1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number

1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x - x) B. x 3 x C. 3x - x D. x - 3x 2) Write the following as an algebraic expression

### Curve Fitting. Before You Begin

Curve Fitting Chapter 16: Curve Fitting Before You Begin Selecting the Active Data Plot When performing linear or nonlinear fitting when the graph window is active, you must make the desired data plot

### MULTIPLE REGRESSION EXAMPLE

MULTIPLE REGRESSION EXAMPLE For a sample of n = 166 college students, the following variables were measured: Y = height X 1 = mother s height ( momheight ) X 2 = father s height ( dadheight ) X 3 = 1 if

### One-Way Analysis of Variance (ANOVA) Example Problem

One-Way Analysis of Variance (ANOVA) Example Problem Introduction Analysis of Variance (ANOVA) is a hypothesis-testing technique used to test the equality of two or more population (or treatment) means

### 2013 MBA Jump Start Program. Statistics Module Part 3

2013 MBA Jump Start Program Module 1: Statistics Thomas Gilbert Part 3 Statistics Module Part 3 Hypothesis Testing (Inference) Regressions 2 1 Making an Investment Decision A researcher in your firm just

### 4. Simple regression. QBUS6840 Predictive Analytics. https://www.otexts.org/fpp/4

4. Simple regression QBUS6840 Predictive Analytics https://www.otexts.org/fpp/4 Outline The simple linear model Least squares estimation Forecasting with regression Non-linear functional forms Regression

### Using R for Linear Regression

Using R for Linear Regression In the following handout words and symbols in bold are R functions and words and symbols in italics are entries supplied by the user; underlined words and symbols are optional

### Regression and Correlation

Regression and Correlation Topics Covered: Dependent and independent variables. Scatter diagram. Correlation coefficient. Linear Regression line. by Dr.I.Namestnikova 1 Introduction Regression analysis

### seven Statistical Analysis with Excel chapter OVERVIEW CHAPTER

seven Statistical Analysis with Excel CHAPTER chapter OVERVIEW 7.1 Introduction 7.2 Understanding Data 7.3 Relationships in Data 7.4 Distributions 7.5 Summary 7.6 Exercises 147 148 CHAPTER 7 Statistical

### 2. What is the general linear model to be used to model linear trend? (Write out the model) = + + + or

Simple and Multiple Regression Analysis Example: Explore the relationships among Month, Adv.\$ and Sales \$: 1. Prepare a scatter plot of these data. The scatter plots for Adv.\$ versus Sales, and Month versus

### Figure 1. An embedded chart on a worksheet.

8. Excel Charts and Analysis ToolPak Charts, also known as graphs, have been an integral part of spreadsheets since the early days of Lotus 1-2-3. Charting features have improved significantly over the

### Section 14 Simple Linear Regression: Introduction to Least Squares Regression

Slide 1 Section 14 Simple Linear Regression: Introduction to Least Squares Regression There are several different measures of statistical association used for understanding the quantitative relationship

### 1. The parameters to be estimated in the simple linear regression model Y=α+βx+ε ε~n(0,σ) are: a) α, β, σ b) α, β, ε c) a, b, s d) ε, 0, σ

STA 3024 Practice Problems Exam 2 NOTE: These are just Practice Problems. This is NOT meant to look just like the test, and it is NOT the only thing that you should study. Make sure you know all the material

### Engineering Problem Solving and Excel. EGN 1006 Introduction to Engineering

Engineering Problem Solving and Excel EGN 1006 Introduction to Engineering Mathematical Solution Procedures Commonly Used in Engineering Analysis Data Analysis Techniques (Statistics) Curve Fitting techniques

### Basic Statistics and Data Analysis for Health Researchers from Foreign Countries

Basic Statistics and Data Analysis for Health Researchers from Foreign Countries Volkert Siersma siersma@sund.ku.dk The Research Unit for General Practice in Copenhagen Dias 1 Content Quantifying association

### Correlation and Simple Linear Regression

Correlation and Simple Linear Regression We are often interested in studying the relationship among variables to determine whether they are associated with one another. When we think that changes in a

### DATA INTERPRETATION AND STATISTICS

PholC60 September 001 DATA INTERPRETATION AND STATISTICS Books A easy and systematic introductory text is Essentials of Medical Statistics by Betty Kirkwood, published by Blackwell at about 14. DESCRIPTIVE

### Example: Credit card default, we may be more interested in predicting the probabilty of a default than classifying individuals as default or not.

Statistical Learning: Chapter 4 Classification 4.1 Introduction Supervised learning with a categorical (Qualitative) response Notation: - Feature vector X, - qualitative response Y, taking values in C