simple regression with spss

8/14/2019 Simple Regression With SPSS

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Example of Using SPSS to Generate a Simple Regression Analysis

Given a desire of a Retail Chain management team to develop a strategy toforecasting annual sales, the following data from a random sample of existing stores has

been gathered:

STORE SQUARE FOOTAGE ANNUAL SALES ($)

1 1726.00 3681.00

2 1642.00 3895.00

3 2816.00 6653.00

4 5555.00 9543.00

5 1292.00 3418.00

6 2208.00 5563.00

7 1313.00 3660.00

8 1102.00 2694.00

9 3151.00 5468.0010 1516.00 2898.00

11 5161.00 10674.00

12 4567.00 7585.00

13 5841.00 11760.00

14 3008.00 4085.00

We can enter the data into SPSSPc by typing it directly into the data editor, or by cuttingand pasting:


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Next, by clicking on Variable View, we can apply variable and value labels whereappropriate:

Assuming, for now, that if a relationship exists between the two variables, it is linear innature, we can generate a simple Scatterplot (or Scatter Diagram) for the data. This isaccomplished with the command sequence:


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Regression Analysis for Site Selection

Simple Scatterplot of Data

Square Footage of Store

70006000500040003000200010000

SalesR

evenueofStore

14000

12000

10000

8000

6000

4000

2000

0

Which yields the following (editable) scatterplot:

We can generate a simple straight line equation from the output resulting when using theEnter Command in regression:


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Variables Entered/Removedb

Square

Footage of

Storea

. Enter

Model

1

VariablesEntered VariablesRemoved Method

All requested variables entered.a.

Dependent Variable: Sales Revenue of Storeb.

Which yields:


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Mode l Summary

.954a .910 .902 936.8500

Model

1

R R Square

Adjusted

R Square

Std. Error of

the Estimate

Predictors: (Constant), Square Footage of Storea.

ANOVAb

1.06E+08 1 106208119.7 121.009 .000a

10532255 12 877687.937

1.17E+08 13

Regression

Residual

Total

Model

1

Sum of

Squares df Mean Square F Sig.



^

So then Yi = 901.247 + 1.686X (noting that no direct interpretation of the Yintercept at 0 Square Footage is possible, sothat the intercept represents the portion ofthe annual sales varying due to factors other

than store size)

and where

SS

SSSS

Coefficientsa

901.247 513.023 1.757 .104 -216.534 2019.027

1.686 .153 .954 11.000 .000 1.352 2.020

(Constant)


Model1

B Std. Error

Unstandardized

Coefficients

Beta

Standardi

zed

Coefficien

ts

t Sig. Lower Bound Upper Bound

95% Confidence Interval for B

Dependent Variable: Sales Revenue of Storea.

b0

b1


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SST = SSR (regression sum of squares) + SSE (error sum of squares)

= sum of the squared differences between each observed value for Yand Y-Bar

SSR = sum of the squared differences between each predicted value of Yand Y-Bar

SSE = sum of the squared differences between each observed value of Yand each predicted value for Y

Coefficient of Determination = SSR/SSt = 0.91 (sample)

Standard Error of the Estimate = SYX = SQRT { SSE / n - 2} = 936.85


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Testing the General Assumptions of Regression and Residual Analysis

1. Normality of Error - similar to the t-test and ANOVA, regression is robust todeparture from the normality of errors around the regression line. This assumption is

often tested by simply plotting the Standardized Residuals (each residual divided byits standard error) on a histogram with a superimposed normal distribution, or on anormalo probability plot. SPSS allows us to perform both functions automatically(while, incidentally, saving the residual values in the original data file if this option istoggled):


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Of course, the assessment of normality by visually scanning the data leaves some

statisticians unsettled; so I usually add an appropriate test of normality conducted on thedata:

Variable n A-D p-value

Stand._Resid. 14 0.348 0.503

Regression Standardized Residual

1.00.500.00-.50-1.00-1.50-2.00

Histogram

Dependent Variable: Sales Revenue of Store

Frequency

5

4

3

2

1

0

Std. Dev = .96

Mean = 0.00

N = 14.00

Normal P-P Plot of Regression Standardized Residual

Dependent Variable: Sales Revenue of Store

Observed Cum Prob

1.00.75.50.250.00

ExpectedCum

Prob

1.00

.75

.50

.25

0.00


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2. Homoscedasticity - the assumption that the variability of data around the regressionline be constant for all values of X. In other words, error must be independent ofX. Generally, this assumption may be tested by plotting the X values against theraw residuals for Y. In SPSS, this must be done by plotting a Scatterplot from thesaved variables:

Click Here


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Results in data automatically added to the data file:


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Then, simply produce the requisite scatterplot as before:

Notice how there is no 'fanning' pattern to the data, implying homoscedasticity.


600050004000300020001000UnstandardizedResidual

2000

1000

0

-1000

-2000


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Other authors, including those who wrote the SPSS routine, choose to plot the X valuesagainst the Studentized Residuals (Standardized Residuals Adjusted for their distancefrom the average X value) rather than the Unstandardized (raw) Residuals. SPSS willgenerate this plot automatically (select this under the Plots panel):

Scatterplot of Studentized Residuals

and Square Footage (X)


600050004000300020001000

StudentizedResidual

1.5

1.0

.5

0.0

-.5

-1.0

-1.5

-2.0

-2.5

Note the equivalence of results between the two plots. Statistically speaking, the X valuesand Residuals may be inferred to be 0.00. We can infer this using the correlation utility inSPSSPc, which tests the null hypothesis that the Pearson rho for the population is equal to0.00:


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It should be noted that the distribution of the data also suggest that an assumption oflinearity is also reasonable at this point.

3) Independence of the Errors - assumes that no autocorrelation is present. Generally,evaluated by plotting the residuals in the order or sequence in which the originaldata were collected. This approach, when meaningful, uses the Durbin-WatsonStatistic and associated Tables of Critical values. SPSS can generate this valuewhen requested as part of the Model Summary:

A number of other statistics are also available in SPSS regarding ResidualAnalysis:

Mode l Summaryb

.954a .910 .902 936.8500 .910 121.009 1 12 .000 2.446

Model

1

R R Square

Adjusted

R Square

Std. Error of

the Estimate

R Square

Change F Change df1 df2 Sig. F Change

Change Statistics

Durbin-W

atson



Correlations

1.000 .000 .015

. 1.000 .959

14 14 14

.000 1.000 .999**

1.000 . .000

14 14 14

.015 .999** 1.000

.959 .000 .

14 14 14

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N

Pearson Correlation

Sig. (2-tailed)

N


Unstandardized Residual

Studentized Residual

Square

Footage

of Store

Unstanda

rdized

Residual

Studentize

d Residual

Correlation is significant at the 0.01 level (2-tailed).**.


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Residuals Statisticsa

2759.3672 10749.96 5826.9286 2858.2959 14

-1.073 1.722 .000 1.000 14

250.7362 512.8126 345.3026 81.3831 14

2771.8208 10518.55 5804.4373 2830.7178 14

-1888.14 1070.6108 -3.25E-13 900.0964 14

-2.015 1.143 .000 .961 14

-2.092 1.288 .011 1.035 14

-2033.82 1442.1392 22.4913 1049.3911 14

-2.512 1.329 -.014 1.111 14

.003 2.967 .929 .901 14

.001 .355 .086 .103 14

.000 .228 .071 .069 14

Predicted Value

Std. Predicted Value

Standard Error of

Predicted Value

Adjusted Predicted Value

Residual

Std. Residual

Stud. Residual

Deleted Residual

Stud. Deleted Residual

Mahal. Distance

Cook's DistanceCentered Leverage Value

Minimum Maximum Mean Std. Deviation N



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Inferences About the Model and Interval Estimates

We can determine the presence of a significant relationship between X and Y by testing todetermine whether the observed slope is significantly greater than 0, the hypothesizedslope of the regression line if no relationship existed. This can be done with a t-test,

which divides the observed slope by the standard error of the slope (supplied by SPSS):

or with an ANOVA model, which provides identical results:

noting that t2, as expected, equals F; and the p-values are therefore equal. Note that SPSSalso provides the confidence interval associated with the slope.

Finally, SPSS allows you to calculate and store both Confidence and Prediction Limitsfor the observed data. After you generate the scatterplot, left double-click on the chart;this will take you to the chart editor:

Coefficientsa

901.247 513.023 1.757 .104 -216.534 2019.027

1.686 .153 .954 11.000 .000 1.352 2.020

(Constant)


Model

1

B Std. Error

Unstandardized

Coefficients

Beta

Standardi

zed

Coefficien

ts

t Sig. Lower Bound Upper Bound

95% Confidence Interval for B


ANOVAb

1.06E+08 1 106208119.7 121.009 .000a

10532255 12 877687.937

1.17E+08 13

Regression

Residual

Total

Model

1

Sum of

Squares df Mean Square F Sig.




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Next:

Then:


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Click on Fit Options


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LCL UCL LPL UPL

3135.52558 4487.50548 1661.27256 5961.75850

2976.95430 4362.80609 1514.25297 5825.50741

5102.73145 6196.07384 3536.24581 7762.55948

9232.70820 11302.74446 7979.09247 12556.36019

2309.22155 3850.24435 897.92860 5261.53731

4028.95209 5219.51308 2497.98206 6750.48311

2349.56701 3880.71656 935.07592 5295.20765

1942.80866 3575.92595 560.87909 4957.85553

5663.35086 6765.16486 4100.00127 8328.51446

2737.79303 4177.06134 1293.06683 5621.78754

8677.59067 10529.18763 7362.03125 11844.74705

7827.42925 9376.22071 6418.64584 10785.004129632.63839 11867.28348 8422.94738 13076.97449

5426.83323 6519.44789 3860.07783 8086.20329

Regression Analysis for Site Selection

Scatterplot of Data Including Confidence & Prediction Limits


600050004000300020001000

SalesRe

venueofStore

12000

10000

8000

6000

4000

2000 Rsq = 0.9098

simple regression with spss

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