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SW318Social WorkStatisticsSlide 1
Regression Analysis
We have previously studied the Pearsons r
correlation coefficient and the r2 coefficient of
determination as measures of association for
evaluating the relationship between an interval level
independent variable and an interval level
dependent variable.
These statistics are components of a broader set of
statistical techniques for evaluating the relationship
between two interval level variables, called
regression analysis (sometimes referred to in
combination as correlation and regression analysis).
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SW318Social WorkStatisticsSlide 3
Elements of Regression Analysis
We will first review previous material on regressionand correlation:
The scatterplot or scattergram
The regression equation
Then, we will examine the statistical evidence to
determine whether or not, the relationships found in
our sample data are applicable to the population
represented by the sample using a hypothesis test.
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SW318Social WorkStatisticsSlide 4
Purpose of Regression Analysis
The purpose of regression analysis is to answer thesame three questions that have been identified as
requirements for understanding the relationships
between variables:
Is there a relationship between the twovariables?
How strong is the relationship?
What is the direction of the relationship?
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SW318Social WorkStatisticsSlide 5
Scatterplots - 1
The relationship between two interval variables can begraphed as a scatterplot or a scatter diagram which showsthe position of all of the cases in an x-y coordinate system.
The independent variable is plotted on the x-axis, or thehorizontal axis.
The dependent variable is plotted on the y-axis, or thevertical axis.
A dot in the body of the chart represented theintersection of the data on the x-axis and the y-axis
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SW318Social WorkStatisticsSlide 6
Scatterplots - 2
The trendline or regression line is plotted on thechart in a contrasting color
The overall pattern of the dots, or data points,succinctly summarizes the nature of the relationship
between the two variables. The clarity of the pattern formed by the dots can be
enhanced by drawing a straight line through thecluster such that the line touches every dot or comes
as close to doing so as possible. This summarizing line is called the regression line.
We will see later how this line is obtained, but fornow, we will look at how it helps us understand the
scatterplot.
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SW318Social WorkStatisticsSlide 7
Scatterplots - 3
The pattern of the points on the scatterplot gives usinformation about the relationship between the variables.The regression line, drawn in red, makes it easier for usto understand the scatterplot.
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SW318Social WorkStatisticsSlide 8
The Uses of Scatterplots
Scatterplots give us information about our threequestions about the relationship between twointerval variables:
Is there a relationship between the two variables?
How strong is the relationship? What is the direction of the relationship?
In addition, the regression line on the scatterplot can
be used to estimate the value of the dependentvariable for any value of the independent variable.
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SW318Social WorkStatisticsSlide 9
Scatterplots: Evidence of a Relationship
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When there is norelationship between twovariables, the regressionline is parallel to thehorizontal axis.
When there is a relationshipbetween two variables, theregression line lies at an angleto the horizontal axis, slopingeither upward or downward.
The angle between the regression line and thehorizontal x-axis provides evidence of a relationship. If
there is no relationship, the regression line will be
parallel to the axis.
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SW318Social WorkStatisticsSlide 10
Scatterplots: Strength of a Relationship
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In this scatterplot, thepoints are very spread outaround the regression line.The relationship is weak.
The spread of the points around
the regression line is narrow,indicating a stronger relationship.
We should check the scale of thevertical axis to make sure thenarrow band is not the result of anexcessively large scale.
The strength of a relationship is indicated by thenarrowness of the band of points spread around the
regression line: the tighter the band, the stronger the
relationship.
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SW318Social WorkStatisticsSlide 11
When the regression line slopes upward to the right,there is a positive, or direct, relationship between the
variables. When the regression line slopes downward,
the relationship is negative, or inverse.
Scatterplots: Direction of Relationship
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In this scatterplot, theregression line slopes upwardto the right, indicating apositive or direct relationship.The values of both variablesincrease and decrease at thesame time.
In this scatterplot, theregression line slopesdonward to the right,indicating a negative orinverse relationship. Thevalues of the variables movein opposite directions.
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SW318Social WorkStatisticsSlide 12
Scatterplots: Predicting Scores
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For any value of the independent variable on thehorizontal x-axis, the predicted value for the dependent
variable will be the corresponding value on the vertical
y-axis.
For the value of theindependent variableon the horizontal axis,we draw a line upwardto the regression line,e.g. 52.
We draw aperpendicular line
from the value on thex-axis to theregression line.
The estimate for the dependentvariable is obtained by drawinga line parallel to the x-axis fromthe regression line to thevertical y-axis and reading thevalue where this line crosses the
y-axis, e.g. 50.
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SW318Social WorkStatisticsSlide 13
The Effect of Scaling on the Scatterplot
The scale used for the verticaly-axis can change the appearanceof the scatterplot and alter ourinterpretation of the strength of therelationship. The three scatterplotson this slide all use the same data. 0
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In this plot, I doubled therange of the y-axis scale to 0to 160, drawing the pointscloser together, and makingthe relationship appearstronger.
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In this plot, I have narrowedthe range of the y-axis scaleto 25 to 75, spreading thepoints, and making therelationship appear weaker.
In the original plot, they-axis is scaled from 0
to 80.
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SW318Social WorkStatisticsSlide 14
The Assumption of Linearity
An underlying assumption of regression analysis isthat the relationship between the variables is linear,meaning that the points in the scatterplot must forma pattern that can be approximated with a straightline.
While we could test the assumption of linearity witha test of statistical significance of the correlationcoefficient, we will make a visual assess tor
scatterplots.
If the scatterplot indicates that the points do notfollow a linear pattern, the techniques of linearcorrelation and regression should not be applied.
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SW318Social WorkStatisticsSlide 15
Examples of Linear Relationships
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FemaleLifeExpectancy
These two scatterplots are for data on poverty of
nations. The plots below show strong linear
relationships. The points are evenly distributed on
either side of the regression line.
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SW318Social WorkStatisticsSlide 16
These scatterplots show a non-linear relationship. The points arenot evenly distributed on eitherside of the regression line. We willoften see a concentration of pointson one side of the regression line
and an absence of points on theother side.
Examples of Non-linear Relationships
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SW318Social WorkStatisticsSlide 17
The Regression Equation
The regression equation is the algebraic formula forthe regression line, which states the mathematicalrelationship between the independent and thedependent variable.
We can use the regression line to estimate the valueof the dependent variable for any value of theindependent variable.
The stronger the relationship between theindependent and dependent variables, the closerthese estimates will come to the actual score thateach case had on the dependent variable.
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SW318Social WorkStatisticsSlide 18
Components of the Regression Equation
The regression equation has two components.
The first component is a number called the y-intercept that defines where the line crosses thevertical y axis.
The second component is called the slope of theline, and is a number that multiplies the value ofthe independent variable.
These two elements are combined in the generalform for the regression equation:
the estimated score on the dependent variable
= the y-intercept + the slope the score on theindependent variable
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SW318Social WorkStatisticsSlide 19
The Standard Form of the Regression Equation
The standard form for the regression equation orformula is:
Y = a + bX
where
Y is the estimated score for the dependentvariable
X is the score for the independent variable
b is the slope of the regression line, or themultiplier of X
a is the intercept, or the point on the vertical axiswhere the regression line crosses the vertical y-
axis
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SW318Social WorkStatisticsSlide 20
Depicting the Regression Equation
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The y-intercept is thepoint on the vertical y-axis where theregression line crossesthe axis, i.e. 1.0.
The slope is the multiplierof x. It is the amount ofchange in y for a changeof one unit in x.
If x changes one unit from2.0 to 3.0, depicted by theblue arrow, y will changeby 0.5 units, from 2.0 to2.5 as depicted by the redarrow.
The regression equation includes both the y-intercept and the slope of the line. The y-intercept is 1.0 and the slope is 0.5.
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SW318Social WorkStatisticsSlide 21
Deriving the Regression Equation
In this plot, none of the points fall on theregression line.
The difference between the actual value for
the dependent variable and the predicted
value for each point is shown by the red
lines. This difference is called the residual,and represents the error between the actual
and predicted values.
The regression equation is computed to
minimize the total amount of error in
predicting values for the dependent variable.The method for deriving the equation is
called the "method of least squares,"
meaning that the regression line minimizes
the sum of the squared residuals, or errors
between actual and predicted values.
y = 0.8 + 0.6 x
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SW318Social WorkStatisticsSlide 22
Interpreting the Regression Equation:the Intercept
The intercept is the point on the vertical axis where
the regression line crosses the axis. It is the
predicted value for the dependent variable when the
independent variable has a value of zero.
This may or may not be useful information depending
on the context of the problem.
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SW318Social WorkStatisticsSlide 23
Interpreting the Regression Equation:the Slope
The slope is interpreted as the amount of change inthe predicted value of the dependent variableassociated with a one unit change in the value of theindependent variable.
If the slope has a negative sign, the direction of therelationship is negative or inverse, meaning that thescores on the two variables move in oppositedirections.
If the slope has a positive sign, the direction of therelationship is positive or direct, meaning that thescores on the two variables move in the same
direction.
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SW318Social WorkStatisticsSlide 24
Interpreting the Regression Equation:when the Slope equals 0
If there is no relationship between two variables, the
slope of the regression line is zero and the regression
line is parallel to the horizontal axis.
A slope of zero means that the predicted value of
the dependent variable will not change, no matter
what value of the independent variable is used.
If there is no relationship, using the regression
equation to predict values of the dependent variable
is no improvement over using the mean of the
dependent variable.
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SW318Social WorkStatisticsSlide 25
Assumptions Required for Utilizinga Regression Equation
The assumptions required for utilizing a regression
equation are the same as the assumptions for the test
of significance of a correlation coefficient.
Both variables are interval level.
Both variables are normally distributed.
The relationship between the two variables is
linear.
The variance of the values of the dependentvariable is uniform for all values of the
independent variable (equality of variance).
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SW318Social WorkStatisticsSlide 26
Assumption of Normality
Strictly speaking, the test requires that the two
variables be bivariate normal, meaning that the
combined distribution of the two variables is normal.
It is usually assumed that the variables are bivariate
normal if each variable is normally distributed, so
this assumption is tested by checking the normality
of each variable.
Each variable will be considered normal if its
skewness and kurtosis statistics fall between 1.0 and
+1.0 or if the sample size is sufficiently large to
apply the Central Limit theorem.
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SW318Social WorkStatisticsSlide 27
Assumption of Linearity
Linearity means that thepattern of the points in ascatterplot form a band,like the pattern in the charton the right:
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When the pattern of thepoints follows a curve, likethe scatterplot on the right,the correlation coefficientwill not accuratelymeasure the relationship.
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SW318Social WorkStatisticsSlide 28
Test of Linearity
The test of linearity is a diagnostic statistical test ofthe null hypothesis that the linear model is anappropriate fit for the data points. The desiredoutcome for this test is to fail to reject the nullhypothesis.
If the probability for the test of statistic is less thanor equal to the level of significance for the problem,we reject the null hypothesis, concluding that thedata is not linear and the Regression Analysis is notappropriate for the relationship between the twovariables.
If the probability for the test of linearity statistic isgreater than the level of significance for the problem,we fail to reject the null hypothesis and conclude
that we satisfy the assumption of linearity.
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SW318Social WorkStatisticsSlide 29
Assumption of Homoscedasticity
Homoscedasticity (equality of variances) means thatthe points are evenly dispersed on either side of theregression line for the linear relationship.
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In this scatterplot, the points extendabout the same distance above and
below the regression line for most of thelength of the regression line.
This scatterplot meets the assumption ofhomoscedasticity.
In this scatterplot, the spread of thepoints around the regression line is
narrower at the left end of the regressionline than at the right end of the regressionline.
This funnel shape is typical of ascatterplot showing violations of theassumption of homoscedasticity.
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SW318Social WorkStatisticsSlide 30
Test of Homoscedasticity
When we compared groups, we used the Levene testof population variances to test for the assumptionthat the group variances were equal.
In order to use this test for the assumption ofhomoscedasity, we will convert the interval levelindependent variable into a dichotomous variablewith low scores in one group and high scores in theother group. We can then compare the variances ofthe two groups derived from the independentvariable.
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SW318Social WorkStatisticsSlide 31
Levene Test of Homogeneity of Variances
The Levene test of equality of population variances
tests whether or not the variances for the two groupsare equal. It is a test of the research hypothesis thatthe variance (dispersion) of the group with low scoresis different from the variance of the group with highscores. The null hypothesis that the variance(dispersion) of both groups are equal.
If the probability of the test statistic is greater than0.05, we do not reject the null hypothesis andconclude that the variances are equal. This is the
desired outcome. If the probability of the test statistic is less than or
equal to 0.05, we conclude the variances aredifferent and the Regression Analysis is not anappropriate test for the relationship between the two
variables.
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SW318Social WorkStatisticsSlide 32
The hypothesis test of r2
The purpose of the hypothesis test of r2 is a test ofthe applicability of our findings to the populationrepresented by the sample.
When we studied association between two intervalvariables, we stated that the Pearson r correlationcoefficient and its square, the coefficient ofdetermination measure the strength of therelationship between two interval variables. Whenthe correlation coefficient and coefficient ofdetermination are zero (0), there is no relationship.
The hypothesis test of r2 is a test of whether or not
r2 is larger than zero in the population.
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SW318Social WorkStatisticsSlide 33
The hypothesis test of r2
The research hypothesis states that r2
is larger thanzero. (a relationship exists)
The null hypothesis states that r2 is equal to zero.(no relationship)
Recall that we interpreted the coefficient ofdetermination r2 as the reduction in errorattributable to with the relationship between thevariables.
The test statistic is an ANOVA F-test which testswhether or not the reduction in error associated withusing the regression equation is really greater thanzero.
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SW318Social WorkStatisticsSlide 34
How the regression ANOVA test works?
We are interested inthe relationshipbetween family sizeand number ofcredit cards.
We will use the sample data we used forcorrelation and regression to examine howthe hypothesis test for r2 works.
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SW318Social WorkStatisticsSlide 35
The scatter diagram or scatterplot
The independent variable isplotted on the x or
horizontal axis.
The dependentvariable is plotted
on the Y orvertical axis.
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SW318Social WorkStatisticsSlide 36
The mean as the best guess
Without taking into accountthe independent variable, ourbest guess for the number ofcredit cards for any subject isthe mean, 7.0.
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SW318Social WorkStatisticsSlide 37
Errors using the mean as estimate
Errors are measured by computing the differencebetween the mean and each Y value, squaringthe differences, and then summing them.
When we compute the answer in SPSS, it will tellus that the total amount of error is 22.0.
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SW318Social WorkStatisticsSlide 38
The regression line
The regression lineminimizes the error(the best fitting or least squares line)
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SW318Social WorkStatisticsSlide 39
The equation for the regression line
SPSS will give us the formula for the regression line in theform Y = a + bX, or for these variables:
Number of Credit Cards = 2.871 + .971 x Family Size
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SW318Social WorkStatisticsSlide 40
PRE reduction in error
Error using mean only (total) 22.000
Error using regression line 5.486
Reduction in error associated with
the regression
16.514
PRE measure (r2) 22.0-5.486 = .751
22.0
SPSS also tells us the amount of error using only the meanand using the regression line.
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SW318Social WorkStatisticsSlide 41
The ANOVA test for the regression
The F statistic is calculated as the ratio of error reduced by regressionsdivided the error remaining.
If the ratio were 1 and these two numbers were the same, we would not
have reduced any error, there would be no relationship, and the p-valuewould not let us reject the null hypothesis.
In this problem, the amount of error reduced by the regression is largerelative to the amount remaining, so the F statistic is large, the p-value(0.005) is smaller than the alpha level of significance, so we reject the
null hypothesis.
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SW318Social WorkStatisticsSlide 42
Interpreting Pearsons r correlationcoefficient
The square root of r2is Pearsonsr, the correlation coefficient.
If we want to characterize thestrength of the relationship, wecompare the size of r to the
interpretive guidelines formeasures of association.
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SW318Social WorkStatisticsSlide 43
Interpreting the direction of the relationship
To interpret the direction of the relationship between thevariables, we look at the coefficient for the independentvariable.
In this example, the coefficient of 0.971 is positive, so wewould interpret this relationship as:
Families with more members had more credit cards.
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SW318Social WorkStatisticsSlide 44
Testing Assumptions in Homework Problems
The process of testing assumptions can easily
overwhelm the task of testing the significance of the
relationship.
Since our emphasis here is testing the hypothesis
that the relationship is generalizable to the
population represented by the sample data, we will
assume that our data satisfies the assumptions
without explicitly testing assumptions.
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SW318Social WorkStatisticsSlide 45
Homework Problem Questions
The question in the homework problems requires us
to look at three things:
Does the hypothesis test support the existence of
a relationship in the population?
Is the strength of the relationship characterized
correctly?
Is the direction of the relationship between the
variables correctly stated?
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SW318Social WorkStatisticsSlide 46
Practice Problem 1
This question asks you to use linear regression to examine the relationship
between [marital] and [age]. Linear regression requires that the dependentvariable and the independent variables be interval. Ordinal variables may beincluded as interval variables if a caution is added to any true findings.
The dependent variable [marital] is nominal level which does not satisfy therequirement for a dependent variable. The independent variable [age] isinterval level, satisfying the requirement for an independent variable.
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SW318Social WorkStatisticsSlide 47
Practice Problem - 2
This question asks you to use linear regression to examine the relationshipbetween [fund] and [attend]. The level of measurement requirements formultiple regression are satisfied: [fund] is ordinal level, and [attend] is ordinallevel.A caution is added because ordinal level variables are included in theanalysis.
Given the assumption that the distributional requirements for linear regressionare satisfied, you can conduct a linear regression using SPSS without examining
distributional assumptions for the variables.
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SW318Social WorkStatisticsSlide 48
Linear Regression Hypothesis Test in SPSS (1)
You can conduct a linearregression using:
Analyze > Regression >Linear
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SW318Social WorkStatisticsSlide 49
Linear Regression Hypothesis Test in SPSS (2)
Move the dependent variableto Dependent: and theindependent variable to
Independent(s): boxes andthen click OK button.
SW318
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SW318Social WorkStatisticsSlide 50
Linear Regression Hypothesis Test in SPSS (3)
Based on the ANOVA table for the linear regression (F(1, 604) = 70.579, p
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SW318Social WorkStatisticsSlide 51
Linear Regression Hypothesis Test in SPSS (4)
Given the significant F-test result, the correlation coefficient (R) can beinterpreted.
The correlation coefficient for the relationship between the independent variableand the dependent variable was 0.323, which would be characterized as a weakrelationship using the rule of thumb that a correlation between 0.0 and 0.20 isvery weak; 0.20 to 0.40 is weak; 0.40 to 0.60 is moderate; 0.60 to 0.80 is
strong; and greater than 0.80 is very strong.
The relationship between the independent variables and the dependent variablewas incorrectly characterized as a moderate relationship. The relationship shouldhave been characterized as a weak relationship.
The answer to the problem is false.
SW318
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SW318Social WorkStatisticsSlide 52
Practice Problem 3
This question asks you to use linear regression to examine the relationship
between [educ] and [age]. [educ] and [age] are interval level, satisfying thelevel of measurement requirements for regression.
Given the assumption that the distributional requirements for linearregression are satisfied, you can conduct a linear regression using SPSSwithout examining distributional characteristics of variables.
SW318
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SW318Social WorkStatisticsSlide 53
Linear Regression Hypothesis Test in SPSS (5)
You can conduct a linearregression using:
Analyze > Regression >Linear
SW318
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SW318Social WorkStatisticsSlide 54
Linear Regression Hypothesis Test in SPSS (6)
Move the dependent variableto Dependent: and theindependent variable toIndependent(s): boxes andthen click OK button.
SW318
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SW318Social WorkStatisticsSlide 55
Linear Regression Hypothesis Test in SPSS (7)
Based on the ANOVA table for the linear regression (F(1, 659) = 9.983, p=0.002),
there was an relationship between the dependent variable "highest year of schoolcompleted" and the independent variable "age". Since the probability of the Fstatistic (p=0.002) was less than or equal to the level of significance (0.05), the nullhypothesis that correlation coefficient (R) was equal to 0 was rejected.
The research hypothesis that there was a relationship between the variables wassupported.
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Linear Regression Hypothesis Test in SPSS (8)
Given the significant F-test result, thecorrelation coefficient (R) can beinterpreted.
The correlation coefficient for therelationship between the independent
variable and the dependent variable was0.122, which can be characterized as a veryweak relationship. .
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Linear Regression Hypothesis Test in SPSS (9)
The b coefficient for the independent variable "age" was -.021,indicating an inverse relationship with the dependent variable.
Higher numeric values for the independent variable "age" [age]are associated with lower numeric values for the dependentvariable "highest year of school completed" [educ].
The statement in the problem that "survey respondents whowere older had completed more years of school" is incorrect.The direction of the relationship is stated incorrectly.
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Practice Problem 4
This question asks you to use linear regression to examine the relationship
between [sei] and [age]. [sei] and [age] are interval level, satisfying thelevel of measurement requirements for regression.
Given the assumption that the distributional requirements for linearregression are satisfied, you can conduct a linear regression using SPSSwithout examining distributional characteristics of variables.
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Linear Regression Hypothesis Test in SPSS (10)
You can conduct a linearregression using:
Analyze > Regression >Linear
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Linear Regression Hypothesis Test in SPSS (11)
Move the dependent variableto Dependent: and theindependent variable toIndependent(s): boxes andthen click OK button.
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Linear Regression Hypothesis Test in SPSS (12)
Based on the ANOVA table for the linear regression (F(1, 629) = .266, p=0.606),there was no relationship between the dependent variable "socioeconomic index" and
the independent variable "age". Since the probability of the F statistic (p=0.606) wasgreater than the level of significance (0.05), the null hypothesis that correlationcoefficient (R) was equal to 0 was not rejected.
The research hypothesis that there was a relationship between the variables was notsupported.
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Steps in solving Linear Regression HypothesisTest Problems - 1
The following is a guide to the decision process for answeringhomework problems about Linear Regression Hypothesis Test
problems:
Are the dependent and
independent variablesordinal or interval level?
Incorrect
application ofa statistic
Yes
No
Make sure that the assumption that the
distributional requirements for linearregression are satisfied is made. Otherwise,you have to check the assumption first.
Our regression problemswill assume that theassumptions are met.
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Steps in solving Linear Regression HypothesisTest Problems - 2
Conduct the linear regression analysis
False
Is the p-value in theANOVA table for the Fratio test
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Steps in solving Linear Regression HypothesisTest Problems - 3
Is the direction of therelationship correctly stated?
Yes
NoFalse
Are either of thevariables ordinal level?
Yes
No
True with caution
True