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OVERVIEW EXAM 3 IS ON DECEMBER 8

GUIDE 1: INTRODUCTION GUIDE 2: CONSTRUCTING A TABLE GUIDE 3: UNIVARIATE STATISTICS AND DISPLAYS GUIDE 4: BIVARIATE BASICS GUIDE 5: BIVARIATE CORRELATIONS GUIDE 6: MULTIVARIATE CROSSTABULATIONS GUIDE 7: BASIC REGRESSION GUIDE 8: REGRESSION SPECIFICS GUIDE 9: SAMPLING

READINGS AND ASSIGNMENTS

 
 

Exam Three is 100 points and should take about one hour to complete. It counts 25 percent toward your final grade. It is the same length as Exam One and Exam Two.

As before, you may be asked to choose the sections of a question that you answer, e.g., select three out of four sections. The purpose of this is to allow you to show off the areas that you know the best. DO NOT answer all choices in such instances. No extra credit! We only grade the first number of designated selections if you answer all the selections in these cases. So what can happen is that (for example, in a 3 out of 4 selection question) you get parts 1, 2 and 4 right, but I only grade parts 1, 2, and 3, and so your credit is lower than if you had simply answered 1, 2 and 4.

The exam is the same general format as Exam 1 and Exam 2: a mix of multiple choice, true-false, short essay, and data interpretation questions. Data interpretation problems similar to Assignments 4 and 5 comprise somewhat over half of Exam 3. You may add a SHORT explanation to any short-answer question.

The data interpretation questions will be comparable to the assignments. You will see two examples below under the SAMPLE QUESTIONS section plus a link to the regression example on Guide 8.

PLEASE BRING AN INEXPENSIVE HAND-HELD CALCULATOR (e.g., a TI 30).
You may end up doing some phis, square roots or squares.

The FOCUS of Exam Three is on the following:

• Three-way crosstabulations with the inclusion of a control variable.
• Causal relationships in three way crosstabulations.
• Basic regression analysis.
• Sampling (a small basic amount).

This exam covers the following in Huff: Chapter 8, pp. 87-99  Chapters 9 and 10, pp. 100-142 (SKIM only) This exam covers these chapters in Agresti and Finlay: Chapter 10, pp. 356-373 Chapter 9, pp 301-342 Chapter 11, pp 382-404 and pp 411-421 OMIT Agresti and Finlay Chapter 2, pp.18-29. This is not required for Exam 3

It also covers:
All lectures
All demonstrations
Course Web sites through Guide 9 (but basic samples only, announced in class)
Aassignments through Assignment 5,
and associated links, including any material (e.g., Exam and Assignment feedback) that I have placed in Blackboard.

You DO NOT have to memorize any formulae for the pdfs; you DO need to know that a pdf (and which one) produces the test statistics that you use to decide whether your results are a sampling accident or whether they are "real". Some of these formulae are reproduced in Guides 3 - 5 in context.

Inevitably, this material is cumulative. For example:

• You must understand bivariate associations in order to understand what happens when you apply a third control variable.
• You need to understand standard deviations, one way analysis of variance, t-tests, bivariate correlations, and variances in order to understand why regression analysis is valuable.
• You should understand the basic logic of "classic" testing for statistical significance, null hypotheses, and alternative hypotheses to solve the data interpretation problems.
• You must understand how causal guidelines for nonexperimental data work for bivariate associations in order to apply this logic to multivariate associations.

Only the basics of sampling will be on the exam and not very much of that.
Do you know the importance of using a probability sample?
Do you know the definition of a probability sample?
Do you know how probability samples differ from non-probability samples?

You won't have complicated formulae to work with. You won't have a formula for B to solve or even one for R². However, I expect you to have examined the fomulae sufficiently so that you know, for example, that R² is the ratio of the explained sum of squares to the total sum of squares (and what the total sum of squares is) or that it is the denominator of B that keeps B in metric units.

You will not have to work any of the formulae in Agresti and Finlay's book. So, for example, you won't have to calculate an r or a Chi-Square or a t-test from scratch. You won't need to look up any probability levels in tables. All numbers will be provided for you, although you may have to calculate a square or square root.

However, you DO need to know what a pdf is and how it creates a sampling distribution. We gauge our results against that hypothetical sampling distribution that is created by your null hypothesis and assumptions about your sample (e.g., obtained through simple random sampling). You will need to understand what the probability levels mean or how to interpret basic regression coefficients IN WORDS.
 

WHAT WILL BE ON THE EXAM

How to "read the results" in a bivariate table. Think of the skills that you needed for Assignments 3 and 4. You will need these as you examine three-way cross-tabulations.

• Reading percentages accurately
• Locating measures of statistical significance
• Locating correlation coefficients
• Distinguishing correlation coefficients from probability levels
• Selecting the best correlation coefficient for your data
• Assessing both statistical AND substantive (practical) significance

Familiarity and comfort with three-way crosstabulations. Understanding multivariate distributions.

Understanding how the results using control variables may alter how we view the causal relationship between the original independent variable and the original dependent variable.

Knowing what the following terms mean in three-way crosstabulations:

• Interaction (conditional, specified, moderated) effect
• Extraneous or direct relationship
• Joint relationship or joint effects
• Intervening or mediated relationship
• Spurious relationship
• Suppressor effect

Understanding the similarities and differences between extraneous and joint relationships.
 

• The intercept
• The slope term(s) (B and BETA)
• The meaning of R² as a "fit" to the regression line and as "percent explained variance"
• An ESTIMATED versus an OBSERVED dependent variable score
• An ESTIMATED versus an OBSERVED regression equation
• Metric B versus Beta Weight coefficients
• Dummy variables
• The standard error of the estimate (and its comparison with the standard deviation of y)

Being able to write out a regression equation, including a NUMERIC regression equation.

Being able to state what each coefficient in a regression equation (including the constant term) does to the dependent variable IN WORDS.

Being able to accurately interpret the many tests of statistical significance in one multiple regression equation, for R² and for each of the Bs. Understanding the null hypotheses for R² and for the Bs.

Assessing which independent variable had the greatest relative influence on the dependent variable.

Being able to describe the strength and direction of each beta weight.

Knowing the difference between statistical significance and substantive (practical) significance--effect size, whether you examine a correlation coefficient, differences among means, or a regression analysis.

Being able to place three variables in a causal chain, if appropriate -- and being able to know when it's NOT appropriate to do so.
 

Understanding why researchers usually take samples rather than studying an entire population.

Being able to distinguish between probability and nonprobability samples.

Being able to accurately define a probability sample.

Understanding the importance of having a comprehensive list of the population under study -- or having a procedure that simulates the creation of such a list.

Naming probability samples and knowing what makes each one a little different.

Understanding why we can only use inference statistics on probability samples from the defined population.

Naming nonprobability samples and being able to recognize them.
 
 

For this exam, our differences are mainly ones of terminology.
 

1. Terminology (I)

I really like the Agresti and Finlay text, and I hope  you will keep it as a reference book. It is an excellent basic textbook.

However, like many statisticians, the use of terminology varies. Agresti and Finlay are particularly apt to employ their own terminology, which is not always the terminology most widely used in the field. To complicate matters further, especially in multivariate analyses, there are often many terms for the same concept that different statisticians may use.

Here is a "glossary"-vocabulary section for review:

EACH ROW STATES BASICALLY EQUIVALENT TERMS

Explained Sum of Squares = Regression Sum of Squares = Model Sum of Squares

Sum of Squared Error =Unexplained Sum of Squares =Error Sum of Squares =Residual Sum of Squares

Intervening relationship = causal chain = indirect causal effect = mediated effect

Joint relationship = multiple causes of the dependent variable or multiple causation

Statistical Interaction = specification or specified = conditional = moderated

 

2. Terminology (II)

Agresti and Finlay use the term statistical control as a generic in multivariate analyses to signify that a control variable (or variables) is being used.

In fact, the term statistical control is usually reserved for regression-type analyses (also sometimes loglinear models which are multivariate models for categorical data) in which the effects of control variables are assessed mathematically through techniques such as linear algebra, and the casebase is analyzed as a whole.

The term physical control is often reserved for multivariate cross-tabular analyses in which the casebase is PHYSICALLY SEPARATED into separate tables in order to assess the bivariate relationship within tables created by each category (value) of the control variable.

REVIEW! 3. "Practical" versus "substantive" significance (Terminology III)

The term "statistical significance" really is an awful term. It implies that if your results are reliably nonzero in the population that they are IMPORTANT! However, in a large sample nearly every association between variables, or differences in means across groups, will be "statistically significant." All that is going on is that with large samples, the standard error of the sampling distribution is very, very small. As a consequence, the results are highly stable from sample to sample. With a tiny standard error, your results are also much more likely to be different from zero  (or any other number that you pick for your null hypothesis), no matter how weak they are.

If your results are statistically significant, but substantively or practically weak or very weak, that is, the effect size is small, you should not call the Tallahassee Democrat to report them.

We should probably call these "statistically significant" kinds of results "statistically stable"--that's a much more descriptive term.

Substantive or practical significance or effect size mean just about the same thing, and you have your choice of which term to use. This term really does refer to how important your results are.

Once you have established that your results are nonzero through some type of inference test, one way to examine effect size is the one I emphasized in the middle section of our course: to begin by assessing the strength of a bivariate correlation. (Usually univariate results receive less attention, unless they are novel or striking.)  Another way to examine practical or substantive importance is to look at the implications of your results. If the implications are strong enough, they may point the way to a new intervention, a new educational method, maybe even a new public policy.

"Statistical significance" is only the BEGINNING of understanding why you found the results you did and what they mean. It is a necessary first step, of course, because you don't want to make a big deal out of essentially zero results, the random fluxuations that can occur from one sample to the next (how embarrassing!) But once you have addressed that criterion, it is time to investigate substantive significance, and, later on, the causal meaning of your association.

And, of course, you should also consider your discipline. Numerically weak results in one scholarly discipline may represent a real advance, depending on the state of knowledge in your discipline or major field. Conversely, results which appear strong may be considered moderate at best in another discipline where the state of knowledge is more advanced.

In small samples, the reverse can happen. You will get results that appear to be moderate, or even strong but that are NOT statistically significant. However, in small samples, the standard errors of the correlation coefficients are relatively large. As a result, the confidence interval for the correlation coefficient may well contain "0". If so, your results are zero in fact in the population, no matter what they appear to be in your particular sample. This is why we FIRST test for statistical significance.
 

Agresti and Finlay show many correlation coefficients in their text, some of which we also use and some that we don't in this class. There are also some we use (Phi/V) or note () that Agresti and Finlay do not address. That's because there are DOZENS of correlation coefficients. Many of them are specialized. Every data analyst has her or his favorites.

I have concentrated on those that (1) are used very frequently, so you are likely to encounter them when you read journals and conference papers; (2) have a PRE interpretation, which is an elegant way to describe the relationship between two variables; and (3) tend to have about the same value in cross tabulation tables.
 
 

5. Multiple Regression

Agresti and Finlay use one chapter for Simple Regression, the next chapter for multivariate crosstabulations, and then another chapter for Multiple Regression. I feel that it is more fruitful and economical to examine simple regression as a "special case" of multiple regression. So many popular press journals (e.g., Newsweek or U.S. News and World Report) now present regression results and graphs that I believe it is part of being an educated consumer of data analysis to understand the basics of multiple regression. Do understand that there are many complexities in regression analysis and regression-type models that are considerably beyond what we can cover in EDF 5400.

At this point, you should be able to comprehend the general material in both Chapters 9 and 11.
 

6. Nice Illustrations in Agresti and Finlay

Check out page 373 and page 421.

HINT: CHECK OUT THIS CLASS WEB SITE FOR THESE TERMS 

BIVARIATE CORRELATIONS.You must understand these to understand multivariate analyses. Both three-way crosstabulations and multiple regression have the bivariate correlation as a building  block.

FIRST, WE STUDIED THE STATISTICAL SIGNIFICANCE OF A RELATIONSHIP BETWEEN TWO VARIABLES.

CHI-SQUARE WAS ONE EXAMPLE.

THE T-TEST WAS A SECOND.

t-tests are also used as the pdf to test the metric B coefficients in regression. That null hypothesis typically is:

H_o : B = 0

Try it: What would be an ALTERNATIVE HYPOTHESIS FOR THE B?

HINT: REVIEW THIS CLASS WEB SITE 

One of the data interpretation problems on your exam (there will be two) requires you to choose a correlation coefficient, just as you did in Assignment 3 and Assignment 4.

What are some WIDELY USED CORRELATION COEFFICIENTS?

HINT: REVIEW THIS CLASS WEB SITE 

How do you know if your correlation coefficient is worth talking about?
Or calling the Tallahassee Democrat about?

HINT: REVIEW THIS CLASS WEB SITE ABOUT COEFFICIENT STRENGTH 

(And, remember, for later on,  you will need to know what is considered typical in YOUR discipline.)

LINEAR, MONOTONIC OR NONLINEAR? How do you know?

HINT: REVIEW THESE CONSTRUCTS HERE: 

DON'T MIX UP THE SAMPLE VALUE OF A CORRELATION WITH THE POSSIBLE POPULATION VALUE. The sample value can fluxuate. The population value will not.

Review the Assignment 3 Feedback spot HERE

DO YOU REMEMBER THE CAUSAL GUIDELINES FOR NONEXPERIMENTAL DATA?

You'll need them to place THREE variables in order.

REVIEW THESE GUIDELINES HERE 

CHECK IT OUT HERE 

AND HERE TOO 

REVIEW BASIC REGRESSION 

REVIEW WHAT PREDICTION OR ESTIMATED EQUATIONS LOOK LIKE HERE 

See the regression example below under SAMPLE QUESTIONS TOO.

REVIEW THE RESIDUALS, A CRITICAL COMPONENT OF REGRESSION ANALYSIS 

HERE'S WHAT MAKES R² "TICK". REVIEW 

USING THE F-TEST 
Remember the null hypothesis here is H₀: F = 0
The alternative is H₀: F > 0 (F cannnot be negative)

AND THE T-TESTS FOR THE BS (the same site as above)
Remember the null hypothesis here is H₀: t = 0 for each B
The alternative is H₀: t =/= 0 (2-tailed or 2-sided test)

And in Guide 8 here: 
The t-test for each B simultaneously tests the statistical significance of each Beta Weight too.

REVIEW THE DIFFERENCE BETWEEN BIAS AND RANDOM ERROR HERE: 

BASIC SAMPLING 

PROBABILITY SAMPLES 

NONPROBABILITY SAMPLES 

EVERYONE MUST TAKE GOOD SAMPLES! 

This is NOT an inclusive list. However, it should serve to give you a sample of the kinds of questions that will be on Exam Three.

Multiple choice. Select the one best or most appropriate alternative response for each question.

1. You controlled using a third variable in a set of crosstabulation tables. In the results, one partial subtable correlation is zero but the other correlation is positive. We call the overall relationship:
 

[  ]A.	Direct
[  ]B.	Interaction
[  ]C.	Spurious
[  ]D.	Intervening

B. The relationship between the original independent and dependent variables IS DIFFERENT across categories of your control variable. This is statistical interaction (often called conditional relationships or specification or moderated relationships).

2. You controlled using a third variable in a set of crosstabulation tables. The values of all the partial subtable correlation coefficients are very similar (within .10) to the original bivariate correlation coefficients. The control variable IS ALSO correlated with the dependent variable. We call the overall relationship:
 

[  ]A.	Interaction
[  ]B.	Intervening
[  ]C.	Joint
[  ]D.	Spurious

C. Both the original independent variable (after controls) and the control variable affect the dependent variable. This is a joint relationship. (Sometimes called multiple causes or multiple determination.)

In an extraneous relationship, although the correlations in  the subtables are similar to the original table for all cases combined, the CONTROL variable is weakly related at best to the dependent variable. Extraneous relationship and poor (statistical) choice of a control variable.

3. You controlled using a third variable in a set of crosstabulation tables. You found that one partial subtable correlation is negative and the other partial correlation is positive. We call the overall relationship:
 

[  ]A.	Direct
[  ]B.	Interaction
[  ]C.	Spurious
[  ]D.	Intervening

B. Again, the correlation between the original independent and dependent variables IS DIFFERENT across categories of your control variable. In this example, it is positive in one case and negative in the other. This is statistical interaction (often called conditional relationships or specification or moderated relationships).

4. After you introduce a control variable, the original bivariate relationship drops to zero. The control variable turns out to cause both the original independent and the original dependent variable. We call the overall relationship:
 

[  ]A.	Direct
[  ]B.	Joint
[  ]C.	Spurious
[  ]D.	Intervening

C. This is a spurious relationship because the control variable causes both of the original variables. It is really as if you had two dependent variables. Numerically, a spurious relationship and an intervening (mediated) relationship look about the same but causally they are quite different.

For students who want to follow up on issues in causality in nonexperimental data, the classic is Morris Rosenberg's 1968 Basic Books volume, The Logic of Survey Analysis.
 

Correlation coefficients:
 

	Eta	Use with one nominal independent variable and one interval-ratio dependent variable. REMEMBER: if you can use a nominal variable, you can use any level so you can use eta with any level independent variable. Useful for nonlinear relationships too.
	Phi	Use with one nominal variable & one nominal (or ordinal) variable REMEMBER! Whatever you can do with nominal data, you can do with ordinal, interval, and ratio data.
or r	Rho	Use with two interval or ratio level variables ONLY (Pearson's r)
-b	Tau-beta	Use with one ordinal & one ordinal (or interval-ratio) variable. ASYMMETRIC (pick an independent variable)
-c	Tau-gamma (Tau-c)	Use with one ordinal & one ordinal (or interval-ratio) variable SYMMETRIC (no independent variable selected)
	Gamma	Use with one ordinal & one ordinal (or interval-ratio) variable (BAD choice, nearly always artificially inflated)

We write the probability (p) of observing a relationship solely by chance as:

p  =      EQUALS                    or
p  <      LESS THAN             or
p  >      GREATER THAN

some figure between 0 and 1.
Probabilities are always between 0 and 1.
Here are some examples:

If there were NO relationship in the population (the correlation in the population is zero)  then:

Please refamiliarize yourself with the < (less than) and > (greater than) signs, remember that it is OK to write LT and GT on Exam 3 if you tend to mix them up (e.g., p LT .001).
 

1. When one variable in an association is nominal and the other variable is ordinal, we typically use nominal measures of association.
      TRUE or     FALSE ?

In MOST cases we drop back to the lower level of data to choose a correlation coefficient.

2. A correlation of -.74 is stronger than a correlation of +.32.

      TRUE     or     FALSE  ?

We disregard the sign and use the absolute value WHEN WE ASSESS THE STRENGTH of a correlation.
We do the same thing with Beta Weights, too. A b* of -.51 is larger than a b* of +.30.

3. In multiple regression, we measure how much ALL of the independent variables PUT TOGETHER predict the dependent variable with the (CHECK ONE:)
 
 

[  ] A.	Metric B
[  ] B.	r
[  ] C.	R2
[  ] D.	t-test

R² is the tool that is used to examine the percent of variance explained in the dependent variable, or the strength of predicting your dependent variable.

Since R² is a fraction, you must multiply it by 100 to get the percent variance explained.
It is called the explained variation because one way to think of R² is the TSS - SSE over the TSS.
(Don't know these terms? Review Guide 7 and Guide 8.)

4. A desirable correlation coefficient has an ERG ratio interpretation.

                           TRUE     or     FALSE  ?

No way! You must have been thinking of PRE (percentage reduction in error) instead.

5. A spurious relationship means you have a true causal association between the original independent variable and the original dependent variable.

                            TRUE     or     FALSE ?

False. A spurious relationship means that the original correlation between the independent and the dependent variable was an illusion. It only occurred because the same REAL independent variable (size of the fire) caused BOTH the original independent variable (number of fire trucks) and the original dependent variable (dollars of fire damage). Diagramatically, it looks like this:
 
 

When in doubt, diagram it out!

A mediated (intervening) relationship looks like this diagram below instead:

Variable 1  Variable 2  Variable 3

For example:

Gender affects choice of science discipline. Choice of science discipline then affects science factual knowledge.

 6. Chi-square is a good measure of correlation.

                             TRUE    or      FALSE ?

Chi-square is NOT a correlation coefficient. Chi-square is a pdf that we use to assess our sample table results. A very large sample chi-square out in the "tail" of the chi-square pdf (under the null hypothesis that X² = 0) suggests that chi-square is non-zero in the population. Chi-square is thus used to test null hypotheses and assess "statistical significance."

7. Causation implies correlation.

      TRUE or     FALSE ?

If one variable causes a second variable, they SHOULD be correlated. However, the reverse does not hold. Correlation does not imply causation.

8. Spurious and explained (intervening) relationships produce different numerical statistical results.

      TRUE     or     FALSE ?

Numerically, spurious and explained relationship produce identical results: the original correlation between the independent and the dependent variable drops substantially from the original bivariate table to the partial correlations within categories of the control variable. Only by drawing causal diagrams and applying causal guidelines for nonexperiment data can you begin to get an idea of what the overall relationship is. For a spurious example, see question 5, immediately above. For another intervening or mediated relationship example see below:

EDUCATIONAL LEVEL & TYPE   TYPE OF JOB   INCOME

Education has an INDIRECT or mediated CAUSAL effect on income because education affects the kind of job that someone has. Type of job mediates the relationship between education and income. This is a REAL causal relationship, not an illusion. You have elaborated the original independent-dependent variable relationship.

REVIEW INTERVENING VERSUS SPURIOUS RELATIONSHIPS HERE 

9. In multiple regression, we measure the net impact of one independent variable on the dependent variable in a prediction equation (controlling other independent variables) with the (CHECK ONE:)
 

[  ]A.	Metric B
[  ]B.	Beta weight
[  ]C.	R2
[  ]D.	r

If you want to predict someone's weight in pounds, or their income, or the number of home computers they own, or their years of longevity, use the metric B. It comes out in units of the dependent variable (e.g., pounds, dollars, home computers or years).

10. In multiple regression, we measure the RELATIVE net impact of an independent variable on the dependent variable within a single equation (controlling other independent variables) with the (CHECK ONE:)
 

[  ]A.	Metric B
[  ]B.	Beta weight
[  ]C.	R2
[  ]D.	r

Use the beta weight to compare the relative net influence of each independent variables on the dependent variable (controlling all the other independent variables). All the beta weights come out in STANDARD DEVIATION UNITS so that you can directly compare them WITHIN A SINGLE EQUATION. (Do NOT compare beta weights ACROSS equations.)
 

For each of the following variable triplets,

• FIRST place the variables in causal order. Draw diagrams to help you if you like.
• SECOND, justify the causal order using our course causal guidelines. Click to review in Guide 5.
• THIRD, indicate if you have an intervening (mediated) or spurious relationship. (Or some other kind of causal relationship if that's the case.)

ANSWERS HERE:

1. Gender precedes both education and occupation in time. For most people, educational level is both earlier in time and also often a necessary condition for type of occupation. So:

Gender     Educational level      Occupational type

This is an INTERVENING or MEDIATED relationship because education INTERVENES between gender and occupation.

2. Race and region of ORIGIN probably have a SYMMETRIC relationship with each other; both are fixed at birth, neither one causes the other. On the other hand,both race and region precede income in time, and both may have some causal relationship, i.e., a JOINT relationship to income (e.g., salaries in the Southern U.S. are lower than in other parts of the country). Causally, here's what we probably have:
 

This relationship is NEITHER SPURIOUS NOR INTERVENING because no variable "comes in between" the original independent and the original dependent variable. (Instead it is probably a JOINT relationship.)

3. Gender (again) is fixed at birth. Given time order, it CANNOT be caused by either the number of science knowledge courses or science knowledge! Logically, low science knowledge will NOT cause someone to take FEWER science courses. However, more science courses are probably a prerequisite for more knowledge. Thus, the causal sequence probably looks like this:

Gender    Number Science Courses      Science Knowledge Score

So causally, the order is gender first, number of science courses as an intervening variable, then science knowledge score.
 

Here are crosstabulation results from  a partial subset of the General Social Survey. The dependent variable is PREMARSX, whether the respondent believes that premarital sex is "wrong" or "not wrong". The independent variable is POLVIEWS, whether the respondent self-identifies as a political liberal, moderate or conservative. Use these data altogether to answer all the questions 1 - 11. First, here are results from the total sample. Use the bivariate results for questions 1 - 8.

THE RELATIONSHIP BETWEEN DEGREE OF POLITICAL CONSERVATISM AND BELIEFS ABOUT PREMARITAL SEX  (General Social Survey subsamples)

1. Is the premarital sex question nominal, ordinal, interval, or ratio?

You can rank the categories so it is ORDINAL. People who think premarital sex is wrong feel so more strongly than those who say "not wrong."

2. Is "degree of conservatism" nominal, ordinal, interval, or ratio?

The "tipoff" is the words "degree of." This implies that those who self-identify as "conservative" are more conservative than self-identified liberals or moderates. You can rank order the categories but can't assign equal interval numbers to them. This variable is ORDINAL.

3. Which variable is the INDEPENDENT VARIABLE: the "premarital sex" question or degree of conservatism? How do you know?

This one invokes the "general to specific" guideline, which is especially appropriate to use when you examine two beliefs or attitudes. Very specific beliefs or attitudes tend to be very unstable and change relatively easily. However, general ideology (such as political liberalism or conservatism or religious values) tend to be quite stable.

4. Are these data (CHECK ONE:)

   [  ]A. Curvilinear
   [  ]B. Linear or
   [X]C. Monotonic
   [  ]D. The data do not allow us to make this decision

BRIEFLY, how do you know?

Both variables are ordinal and self-identified political views has more than two categories so we can begin to address this question. The percentage saying "wrong" increases steadily as the respondent becomes more conservative, but at an irregular rate.

5. Which correlation coefficient is best to use for these data?

[  ]A. Chi-square
[  ]B. Phi
[  ]C. r
[  ]D. Tau-beta

BRIEFLY, why did you choose this particular correlation coefficient?

Chi-square isn't a correlation coefficient, r is for interval data, you could use Phi, but since you have two ordinal variables and a probable asymmetric relationship, Tau-b is the best choice.

6. (2) Was the correlation coefficient you chose statistically significant? CHECK:

   [ X ]YES or   [   ]NO

What was the probability level?
 

7. Was the strength of this coefficient (CHECK ONE):

[  ]A. Very weak
[X]B. Weak
[  ]C. Moderate
[  ]D. Strong or
[  ]E. Very strong?
[  ]F. The data do not allow us to make this decision

According to our chart, the Tau-b of 0.15 is WEAK. Recall that different disciplines may use different designations depending on (1) the state of knowledge in the discipline and (2) the use of relatively homogeneous versus relatively heterogeneous samples.

8. BRIEFLY describe how attitude toward premarital sex correlates with general political orientation (i.e., what the results mean in words):

People who self-identify as political moderates or conservatives are more likely to say premarital sex is wrong than people who self-identify as political liberals.

Now, here are the results from the association between self-identified political views and attitude toward premarital sex broken down separately for men and women. Use these data to answer questions 9 - 11.

THE RELATIONSHIP BETWEEN DEGREE OF POLITICAL CONSERVATISM AND BELIEFS ABOUT PREMARITAL SEX FOR:

THE RELATIONSHIP BETWEEN DEGREE OF POLITICAL CONSERVATISM AND BELIEFS ABOUT PREMARITAL SEX FOR:

9. Was your chosen correlation coefficient statistically significant:

      FOR MEN:   [  ] YES   or  [X] NO

          FOR  WOMEN:  [X] YES  or  [  ] NO
 
 

10. What was the strength numerically and in words of your chosen correlation coefficient:

    FOR MEN:        numerically = 0.08 strength=zero (it's not statistically different from zero)
    FOR  WOMEN: numerically = 0.22 strength=weak

11. Altogether, what kind of causal relationship do you have in the data among POLVIEWS, PREMARSX and GENDER? CHECK ONE:

[  ]A. Direct (extraneous)
[X]B. Interaction (Specification/Conditional/moderated)
[  ]C. Intervening (Indirect/Elaborated/explained/mediated)
[  ]C. Joint
[  ]D. Spurious

BRIEFLY explain the rationale behind your choice:
 

REVIEW THE SECTION ON INTERACTION EFFECTS HERE 
 
 

This example is a variation on scores on the science 10 item quiz. You will recall that women scored lower than men on the basic science quiz, even with controls for educational degree level and the number of college science courses.

One possible reason why this sex difference in science quiz scores occurs may not be because women get more wrong answers, but because women are more willing than men on general public opinion surveys to admit that they don't know the answer -- or are not sure, whereas (in general) men are more willing to guess at the answers. For example, a 2003 article in the journal Public Perspective found that women were more willing to say "I don't know" to a series of political knowledge questions than men were.

If, in fact, there is a sex difference in willing to say one doesn't know the answer, this should be considered when we assess any sex differences in science knowledge scores. Once again, I will also use educational degree level and number of college science courses as additional independent variables. We can call this dependent variable uncertainty about science knowledge or the "I don't know" score. It is the sum of "I don't know" answers over the 10 basic science knowledge items.

FIRST, I look at the bivariate sex difference on uncertainty about science knowledge:

"I DON'T KNOW SCORES" ON SCIENCE KNOWLEDGE

t-test ₍₁₈₈₀₎ =  -10.75  p < .001   Eta = 0.241

Notice that on the average, women answer "I don't know" on almost twice as many science questions as men (1.72 versus 0.97). Although this difference was highly statistically significant (there were 1882 cases in the 1999 survey), it was weak in strength.

It is fine to use an ANOVA here as a beginning bivariate analysis. I could use nominal, ordinal, interval OR ratio variables as my independent variable (my dependent variable must be interval-ratio and it is: "the number of 'I don't know' items"). Educational level has only four categories, so it is easy to present the results. I can also easily check for any departures from a linear or monotonic relationship this way. If I have a distinctly nonlinear relationship between educational level and the science quiz, I would probably want to rethink using degree level as a predictor for a regression analysis.

"I DON'T KNOW SCORES" ON SCIENCE KNOWLEDGE BY EDUCATIONAL LEVEL
 

GROUP	Mean Score (out of 10)	Standard Deviation
Less Than High School	2.25	2.05
High School Degree	1.54	1.57
College Degree	0.87	1.16
Advanced Degree	0.75	1.12
TOTAL	1.38	1.56

F-test _(3,1878) = 52.03 p < .001   Eta = 0.28

Again, the difference in science knowledge scores across educational levels is highly statistically significant. This relationship is moderate (but really about the same as the effect of gender, 0.28 versus 0.24). Scores decrease at a constant rate per level (about -0.70 points) from the less than high school group to the college degree group, then basically plateau, so the relationship approximates a straight line for the lower educational levels. Better educated respondents answer "I don't know" to fewer questions than poorly educated persons do. Thus, this is a monotonic relationship (that is approximately linear for two out of the three educational jumps).

Once again, the standard deviation on our dependent variable, uncertainly about science knowledge, gets smaller and smaller with each successive educational level (look at the little gray arrows in the table above). The standard deviation is nearly twice as great for those with less than a high school education as it is for those with an advanced degree.

This is another example of HETEROSCEDASTICITY, or UNEQUAL VARIANCES ON THE DEPENDENT VARIABLE ACROSS CATEGORIES OF THE INDEPENDENT VARIABLE. When you have heteroscedasticity, the standard errors of the Bs on your computer output often are inaccurate. Worse yet, the standard errors on the Bs that your computer output presents to you can appear SMALLER THAN THEY REALLY ARE! If this happens, your t-test will be too big, and you might conclude that you have statistically significant findings when you really don't.

While we will forge ahead with this example for illustrative purposes, be aware that you will often need to use something called "Weighted Least Squares" to correct for these unequal variances. Sometimes you will see this in programs as the "generalized least squares" solution where the weight is the inverse of the regression equation.

Next, I ran the zero-order or bivariate correlations among all the variables that I was examining in this analysis. Gender is coded as a dummy variable with female = 1 and male = 0. These are Pearson's r correlations and those are the type of correlations that regression  programs calculate for you. The correlations among these variables are presented below in a correlation matrix.

Since the zero-order or bivariate correlation matrix is symmetric, that is, the top right hand side is the same as the lower left hand side, we will just use the lower left side of the matrix as you see below. There are "1"s on the diagonal of the matrix because the correlation of a variable with itself equals 1.

ZERO ORDER (bivariate) CORRELATIONS AMONG ALL VARIABLES IN THE REGRESSION EQUATION

All correlations are statistically significant at the p < .001 level
(I checked these out with a separate test)

From the correlation matrix, we can see that gender is weakly and positively correlated with saying "I don't know" on the science knowledge items. Women say "I don't know" to more items than men do.

The number of college science classes has a moderate negative relationship (r =- 0.29) with uncertainly about the science knowledge items.

Educational level is moderately and negatively related (r =- 0.27) with uncertainly about the science knowledge items.

In terms of possible multicolinearity, gender has weak negative correlations with the number of science courses and educational level, no problem there. The number of college science classes is strongly and positively related to educational level (r = 0.59) so it MIGHT bear watching later on.

Now, here's the regression of gender, the number of science courses, and educational level on the science knowledge score. You are seeing the output here, hence the row of zeros for the p-level.
 

R = 0.365  R² = 0.13    n = 1882
F_3,1878 = 96.38 p = .0000
Standard error of the estimate = 1.46
Standard deviation of the "I don't know" score = 1.56
 

Given that each B is statistically significant, let's see what each one means IN WORDS.

The B for Gender was 0.61. Given that female = 1 and male = 0, this means that controlling educational level and the number of science courses, women answered "I don't know" on 0.61 more science items than men did. The original sex difference was 0.75 "I don't know" items. The 0.61 CONTROLLED sex difference is nearly as great as the original, uncontrolled sex difference. This means that even after controlling degree level and exposure to science courses, women still say "I don't know" on the science quiz more often.

The B for the number of college science classes was -0.08.This means that for each additional college science class the person took, he or she scored .08 FEWER I don't know answers on the 10 point science quiz. Thus, if we compared someone with 0 science courses with someone who had 10 science courses, the person with 10 science courses would have about 1 "I don't know" answer less (10 * - .08 = -0.80 items) than the person with no science courses at all  (controlling gender and degree level).

The B for educational level was -0.26. For each jump in degree level, the person said "I don't know" on about one-quarter fewer answers. If we compare the jump from less than high school to an advanced degree, the person with an advanced degree on the average said "I don't know" on one item less of the 10 items (3 X -0.26 = 0.78) than someone who never completed high school (controlling gender and the number of college science classes). That's about a 10 percent difference on the quiz scale between someone with a relatively low educational level and someone with a very high level.

Finally, the constant term is 1.84.  If someone were male, never had any education, and never had a college science course, they would answer "I don't know" on nearly 20 percent of the answers (1.84 out of 10 questions).

Thus, the uncertainty about science knowledge score drops with more education, especially more science courses. However, a gender gap remains even controlling those two variables.
 
 

You can present the numeric results in a simple chart like this one:

Number of "I don't know" responses to the science knowledge items
 

The numeric regression equation is:

Y = 1.84 + 0.61 D₁ - 0.08  X₁ - 0.26  X₂    where
Y is the number of "I don't know" answers to the basic science quiz
D₁ is "female" where 1 = female and 0 = male
X₁ is the number of college science classes and
X₂ is the respondent's educational level
 

So, for example, a male (D₁  =  0) with ONE science course (X₁ = 1) and a B.A. degree (X₂  = 3) would answer "I don't know" to slightly under one question on the 10 item quiz:

1.84 + 0 - (.08 * 1) - (.26 * 3) = 0.98

The constant disappears because it is zero in a standardized regression equation.

All the beta weights are of similar magnitude (something you could not tell from the metric regression equation) in terms of their impact on science uncertainty. Educational level has a slightly smaller NET impact than gender or science courses, but not very much smaller; all the Betas are weak in strength.

The Standard Error of the Regression (that is, the average deviation around the regression line for science uncertainty) was  1.46. This is after controlling gender, educational level, and the number of science courses.

The actual standard deviation of the "I don't know" score was 1.56.

You can see that you don't gain very much in predicting average science uncertainty by knowing scores on gender, educational level, and the number of science courses. Although you are a bit more precise in prediction using these three independent variables.

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FIRST study your univariate and bivariate statistics: the means, standard deviations, and the correlation coefficients.
Note any unusually large or small correlations.
MAKE SURE YOU KNOW WHAT THE METRIC IS (pounds of weight? number of household computers? number of library books?) OF YOUR DEPENDENT VARIABLE. This will be the metric you will use for the Bs.

SECOND see if the overall R² is statistically significant. Use the Global F-Test results and look at the "P" for probability level.

If the R² is basically 0 (p > .05), any apparent influence of the predictors on the dependent variable is a SAMPLING ACCIDENT. STOP HERE IN THIS CASE! GO NO FURTHER!

The null hypothesis, H_o : R² = 0

The alternative hypothesis is  H_A:  R² > 0.

Because R² is a squared measure, it cannot be a negative number.

If the significance level for the F test is small (p < .05), then the R² is REAL (non-zero).
Usually this means at least one B is non-zero.
Go to step 3.

THIRD see if the STRENGTH of R² is at least weak (.11 plus). THIS IS R-SQUARE (not R).

If yes, continue to step 4.
If R² is smaller than .11, your results are real but probably not practically important.
Interpret any Bs with extreme caution.

(NOTE: 10% explained variation MIGHT be a big deal, depending on the state of knowledge in your discipline of study. So be sure to interpret the strength of R² with your discipline in mind. Also consider the source of the data. For example, general public samples typically are much more variable than student samples because they are more heterogenous with respect to age, work experience, etc. and there is more measurement error.)

FOURTH NOW examine each of the Bs.
Any B less than twice its own standard error will usually have a significance level greater than .05.
This means any apparent influence of that B is a sampling ACCIDENT and that B is really 0.

Use a marker to note the Bs with statistical significance < .05.
These Bs are REAL or nonzero. Of course, substantively they may be quite small. They are just not zero.

Discuss how the statistically significant Bs raise or lower scores on the dependent variable (in pounds of weight for my example: For example, for each 15 minute period a woman exercised, she would weigh 1 pound less.)

When you describe the B in words, you MUST use the metric in your description (e.g., years of education or number of "I don't know" answers).

It is generally inappropriate and invalid to treat the Bs as if they were correlation coefficients.

CLICK HERE TO REVIEW THE WEIGHT EXAMPLE.

FIFTH Look at the BETA weights of the SIGNIFICANT Bs. (Remember that the Bs that were not statistically significant are really 0 in the population and their corresponding Beta Weights are zero too.)

Rank the Beta Weights from most to least important in terms of absolute value size.
Discuss the strength and direction of each statistically significant beta weight.

For each of the following statements check whether the statement is true or false.