EXAM 2 IS ON
NOVEMBER 3
GUIDE 2: CONSTRUCTING A TABLE
GUIDE 3: UNIVARIATE STATISTICS AND DISPLAYS
NOTE: I composed this sequence in response to a question and replicate it here for all:
CHOOSING A CORRELATION COEFFICIENT: POINTS TO CONSIDER: When deciding on a correlation coefficient, the first thing you look at MUST be the level of data because that could rule certain correlation coeffients out.
I would make relationship form the next thing. You don't want to use a tau or an r when the relationship is not linear or non-monotonic even if all variables are numeric. Tau or r will drastically underestimate the strength of the correlation in non-linear relationships between two variables.
Then I would look at symmetry-asymmetry. If the dependent variable is numeric and has a lot of values that could argue for eta or r (depending on the level of the independent variable and form) because they are more concise.
Go for a PRE coefficient in either its squared or regular version (NOTE: it will be easier to compare effects in the non-squared version).
A directional coefficient
is more informative if all variables are at least ordinal and form is at
least approximately monotonic.
NOTE POSTED TUESDAY 11-2
11:53 AM
EDF 5400 INTRODUCTORY STATISTICS FALL 2004 GUIDE TO THE MATERIAL EXAM 2 DR SUSAN CAROL
LOSH
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PLUS
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ASSIGNMENT 2 FEEDBACK EXAM 1 FEEDBACK ASSIGNMENT 3 FEEDBACK |
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Exam Two is 100 points and should take about one hour to complete. It counts 25 percent toward your final grade.
The exam is in our classroom, regular time, CLOSED BOOK, CLOSED NOTE.
In some cases you will 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 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 One: a mix of multiple choice, true-false, short essay, and data interpretation questions. You may add a SHORT explanation to any short-answer question.
The data interpretation questions will be comparable to the assignments. You will see an example below under the SAMPLE QUESTIONS section.
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PLEASE BRING AN INEXPENSIVE HAND-HELD
CALCULATOR (e.g., a TI 30).
You may have to calculate
percentages or an eta.
This exam covers these chapters and
pages in Darrell Huff, and in Agresti and Finlay:
| Darrell Huff:
Chapter 7, pp. 74-86 Chapter 8, pp. 87-99 Agresti and Finlay:
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It also covers all lectures, demonstrations, and course Web sites through Guide 5, Assignment 3 and Assignment 3 Feedback, and associated links, including any material (e.g., Exam 1 and Assignment feedback) I placed in Blackboard.
Please note that cumulative percentages
and group comparisons may ALSO be covered on this exam, especially material
that showed some gaps in understanding on Exam 1.
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You WON'T have to do any complicated calculations, including:
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I expect you to know
I expect you to be able to identify all the
"pieces" of a table, such as:
How to "read the results" in a bivariate table. Think of the skills that
you needed for Assignment 3.
Understanding the similarities between what a t-test (or, more generically,
a difference of means test) and a bivariate table are trying to do.
Understanding the relationship between an F-test and a t-test (square root of F when the nominal independent variable has only two values).
Understanding the difference between what a difference of means test can do (one at least nominal variable, one interval or ratio variable) and why a t-test or F-test can make a somewhat stronger and more precise statement about the relationship between two variables compared with a bivariate cross-tabulation table, especially if the dependent interval-ratio variable has many values.
Understanding what most null hypotheses for the relationship between two
variables "look like."
Understanding what the different alternative hypotheses for the relationship between two variables "look like," including a "one-tailed test" and a "two-tailed test."
Understanding the null hypothesis and the alternative hypothesis for a difference of means test.
Understanding the null hypothesis and the alternative hypothesis for:
Understanding how levels of statistical significance relate to the results
you would expect in an infinite sampling distribution (or a set of samples
of the same size and type taken at about the same time.)
That results that are "statistically significant" are so unusual under the null hypothesis for the population that you would only expect such extreme results in a sample by chance a tiny fraction of the time.
Understanding that results in a sample could look as if there
is a substantial relationship between two variables when, in fact, your
results are a sampling accident and the relationship in the population
is really zero.
This is because we normally expect some variation in results from sample to sample. The smaller the sample, the larger the variation across samples. Results often (depending on sample size) need to be very large or extreme in order to be different from zero and NOT reflect sampling variations.
Understanding that under MOST null hypotheses, sample results that occur
with a high probability level (close to 1) are probably a sampling
ACCIDENT. You would expect results with a high probability in a large number
of samples if the population association between two variables is really
zero.
Understanding that under
MOST
null hypotheses, sample results that occur with a very small probability
level (.05 or less) are probably "REAL"--that
is the relationship in the population is probably NONzero.
Very
large or extreme results are rare events and occur in a tiny fraction of
samples in the population if that association is truly zero.
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You may have drawn one of the rare samples that produced very strong results, although there is no association in the actual population.
The nice thing about inferential statistics, and what is called TYPE ONE error is that you know your chances of being wrong! They are 5 in 100 or 1 in 1000, or the level of the "p" or probability level in your computer output.
Of
course, accepting the null hypothesis (or failing to reject it) has its
own problems: you may be wrong there too. Your population association is
really nonzero but you accepted a false null hypothesis that the population
association really was zero. We call this a TYPE TWO (or "Beta") error.
Instead of calling them "Type I" and
"Type II" error, I find it easier to keep these two types of possible mistakes
straight if I call them "First Error" and "Second Error." Recall my little
mnemonic chart immediately below to see why:
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| FIRST ERROR ("type one") | Rejecting a true null hypothesis |
| SECOND ERROR ("type two") | Accepting a false null hypothesis |
By the way, Agresti and Finlay are totally correct when they say we don't "accept the null hypothesis" (for example, your sample size may be too small to calculate a stable percentage).
We either reject Ho or fail to reject Ho.
However, as a beginner, I had a very hard time keeping Type I and Type II error straight. The mnemonic I came up with in the table above helped me to remember which kind of error was which. Feel free to use or discard this table depending on how easy it is for you to memorize what Type I and Type II error are.
You
need to be able to examine the statistics for a bivariate table and first
decide whether the results are:
If you have established that the correlation that you observed in your
sample is probably nonzero in the population, you now must address
the substance of that correlation between two variables.
How
strong is "strong" and how weak is "weak"? Can you interpret
the numeric size of the correlation coefficients in your data?
Have you memorized the strength chart
in Guide Five? It is only SEVEN sets of numbers! You need to memorize it
for this semester. Most researchers will agree about the extreme high and
low correlations. Learn what is "normative" in your discipline or
field of study.
REVIEW
THE CHART HERE
Why
are both statistical significance and
substantive
(practical) significance important to assess in a correlation coefficient?
Do you understand how effect size relates
to substantive significance?
What
are some of the similarities and differences between a correlation coefficient
and a difference of means test?
Can
you tell the difference between mere covariation or correlation (symmetric)
and at least being able to designate an independent and dependent variable
(asymmetric)?
Can
you match up the guidelines to help establish causality in non-experimental
(observational) data with actual examples of variables? You
will have some sample questions below AND on Exam 2.
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Probably the biggest difference
I have with the text is that Agresti and Finlay like Gamma and I dislike
it intensely! They are correct that it is a PRE measure and easy to calculate.
They even mention that 
produces
values that tend to be larger than, say, Kendall's 
b
(by the way, a and b
ARE asymmetic; it matters which variable you choose as the independent
variable).
Because it is inflated, I think Gamma
is a bad measure to use and recommend against it as misleading so I URGE
you not to choose it (in this course, anyway).
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The other, sometimes sizable, difference I have with our text is they tend to designate their own terms. I agree with their reasoning most of the time, by the way (and I won't count off if you use their terms) but I am trying to use terms consistent with the way in which at least 90 percent of the statistical world uses them. That's why I use the term statistical significance although I think it is a very undescriptive term (see below).
So in the table below, I synopsize the terms I have used in class and the Agresti and Finlay terms:
| CLASS TERM | AGRESTI AND FINLAY TERM |
| Crosstabulation table | Contingency table (either term is OK) |
| Independent variable (much more often used) | Explanatory variable |
| Dependent variable (much more often used) | Response variable (rarely used except by Agresti) |
| No association or no relationship | Statistical independence (each term has advantages) |
| Agreement Pair (slightly more often used) | Concordant pair (either term is fine) |
| Disagreement Pair (slightly more often used) | Discordant pair (either term is fine) |
| two-tailed test (much more often used) | two-sided test (but probably more descriptive) |
| one-tailed test (much more often used) | one-sided test (but probably more descriptive) |
| Substantive significance (I use both, prefer this one) | Practical significance |
Although Agresti and Finlay
use slightly different calculations for the expected frequencies in a Chi-square
you will note the material in Guide 4 actually follows their explanations
of statistical independence very closely and the results should be equivalent.
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A more conventional statistics course might have you actually working computational formulas and doing calculations. Agresti and Finlay have problems like this in each chapter. Ironically, I am both more interested in the true mathematical statistics probability density functions and the actual data analysis than I am in computational formulae. I want you to get some notion of the mathematics that produce pdf curves. The actual probabilities we use in "significance testing" operate off the areas under these curves. Extreme results are typically found in the "tails" of these distributions. In other words, I want you to have some idea of where these distributions come from. That's why some of the pdf formulaes have been inserted into our online guides.
The key to remember, however, is that you, and the researchers whose writings you read, work with actual sample data. What they (and you) should be doing is assessing how well their actual data match the assumptions in the probability density functions (the normal, the t, the F, Chi-square) that are available for us to use. Those assumptions, and how well they are met, tell us which pdfs we can "borrow" from to test our inferential statistics with.
On the other hand, I want
you to become familiar with what data analysis looks like in practice.
The exercises that you do with your computer assignments are similar to
what a researcher does when s/he analyzes data for a research report. The
mistakes that you have made in your initial forays into this analytic world
I hope will sensitize you--because even more experienced researchers will
make the same mistakes that you do. Many statistics programs are complex
and all of them typically give you far more information than you need or
will use. It is the job of the analysand to selectively use the information
that computers can deliver to us so rapidly.
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The term "statistical significance" really is an awful term. It implies that if your results are reliably different from zero in the population that they are IMPORTANT! As we have already seen, in a large sample many associations 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 for your chosen statistic 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 could pick for your null hypothesis), no matter how weak they are. This is one reason why many statisticians place so much emphasis on "Effect Size" (ES). If the a priori determined effect size is not achieved, no matter how "statistically significant" your results are, they are not what you would call practically important.
You would not call the Tallahassee Democrat to report them. You would not call your family in China to discuss them.
We should probably call these "statistically significant" kinds of results "statistically stable"--that's a much more descriptive term. But to be consistent with most texts and statisticians, "statistically significant" it will be.
Substantive or practical significance 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.
Another way, which I emphasized in this middle section of our course, is to begin by assessing the strength of a bivariate correlation. (Usually univariate results receive less attention, unless they are novel or striking.) Later on, you will want to see if this correlation maintains its strength even when you control for other variables (each control variable essentially represents an alternative explanation for why you found the bivariate results that you did). Another way to examine practical or substantive importance is to look at the implications of your results. If the results 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 random fluctuations 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.
The issue of "effect size" is closely related. You tend to see the term more often when researchers discuss differences of means tests. But the fundamental idea is analagous to substantive significance. Typically you will pick some fraction or multiple of the standard deviation of the mean difference*. If the difference across group means is not at least as large as this a priori criterion, you will not view the result as substantively or practically significant.
*This is different from the standard deviation of a univariate mean. It is the standard deviation of the difference of the means across groups.
(As you can see, Agresti
and Finlay and I basically agree, but I spend more time on the concept
than they do.)
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We concentrate a lot in this section of the course on LOOKING AT DATA, especially TABULAR DISPLAYS. Bivariate (and multivariate) tables are one of the most common analytic and presentational techniques in the human sciences. I have concentrated somewhat more on tables--the "pieces" of a table, interpreting percentages, what the form of a relationship looks like in a tabular display--than Agresti and Finlay do in the text. I think illustrations of the form of a relationship are critical, otherwise, for exampe, you might use a linear or monotonic measure such as Pearson's r or a Tau when you have a curvilinear relationship--thereby UNDERestimating the strength of the relationship.
In turn, Agresti and Finlay presented several more correlation coefficients and group differences techniques for nominal and ordinal data that we will not examine in this course, although these analytic methods can address specialized problems and be quite useful. I hope you will keep your text as a reference for your data analysis because one of these coefficients that we did not examine may just "fit the bill" if you later run into problems with your analysis.
I feel that understanding tables (univariate, bivariate, multivariate) is CRITICAL to making you an educated consumer. Tables are everywhere: in professional reports and journals, in newspapers, magazines, online and presented on TV. A lot of times they are MISpresented too. You need to be able to thread your way through reading a table and interpreting the statistics that accompany it. The strength of an association between two variables is often overlooked in the "race for statistical significance," yet you must look for measures of strength and interpretation of correlation coefficient results to see how important the results really are. And these topics are exactly what this middle section of our course addresses.
This means that a lot of the tabular material in Agresti and Finlay's chapter 8 should come as a review!
Notice, by the way, that
we DO cover many of the same topics (with pretty much the same conclusions)
as the Agresti and Finlay text, although the order in which we examined
them differed somewhat. We examine null and alternative hypotheses, the
sequence of making decisions about findings, and many of the same correlation
coefficients So regardless of which source
you examine, you should be able to understand when results are probably
a random sample fluctuation and when something "real" is going on in the
population. You should have a healthy appreciation for the range of correlation
coefficients and tests of group differences that are available, and when
these are appropriate or invalid to use.
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Agresti and Finlay only mention
Cramer's V in passing (and I didn't see a reference to Phi in our readings,
although I may have missed it.) I guess I like Phi for the same reasons
Agresti and Finlay like Gamma: it's relatively easy to calculate if your
computer program doesn't provide it (SPSS does); Phi-square is also PRE;
and in a two by two table, Phi = b
= Pearson's r. I tend to like PRE correlation coefficients (so do Agresti
and Finlay) and Phi is easier to calculate and generally more available
than Lambda as a measure of association for nominal variables.SDA calculates
Lambda but I have run into problems with the SPSS version.
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I spend a lot more time in general on causal issues than Agresti and Finlay do in your assigned readings.
Partially, this is because I want to "prepare" you for upcoming issues in multivariate analysis, such as regression, when causality becomes a critical issue. If you go on to take a course in structural equation models, for example, being able to establish proposed causal relations BEFORE you analyze your data is critical.
Partially this is because
too many writers (researchers AND journalists) draw unwarranted assumptions
from bivariate correlations.Remember one of my goals for you is to have
you emerge from this course as a more educated consumer.
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LEVELS OF VARIABLE MEASUREMENT is one of the most basic--and most important--areas in data analysis. As you now realize, if you don't know what kind of data you have, you won't know what to do with it or how to interpret the results.
HINT: CHECK OUT THIS CLASS
WEB SITE FOR A REVIEW 
Remember, too, that we establish
the level of measurement conceptually in terms of properties of the particular
variable category system. Generally we do NOT use actual sample distributions
to decide what kind of data we have because particular value distributions
can change from sample to sample.
HINT: CHECK OUT THIS CLASS
WEB SITE FOR THESE TERMS
Review BASICS
ON STATISTICAL INFERENCE
This includes:
Review it here, including examples:
CHECK OUT THIS CLASS
WEB SITE FOR THIS TERM
CHECK THEM OUT ON THIS
CLASS WEB SITE
What are CHARACTERISTICS OF A GOOD CORRELATION COEFFICIENT?
HINT: CHECK OUT THIS CLASS
WEB SITE
NOTE: See more about Eta under the Pearson's r section too. It is a terrific coefficient when you have any level measure of independent variable, and your dependent variable MUST be numeric--interval or ratio.
HINT: CHECK OUT THIS CLASS
WEB SITE FOR THESE TERMS
HINT: CHECK OUT THIS CLASS
WEB SITE FOR CORRELATION COEFFICIENT STRENGTH
HINT: CHECK OUT THIS CLASS
WEB SITE FOR THESE TERMS
Review the Assignment 3 Feedback spot HERE
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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 Two.
Multiple choice. Select the one best or most appropriate alternative response for each question.
If you do a comparison of means computer run, which of the following statistics does the SDA system calculate for you?
[
]A. A chi-square
[ ]B. An F-ratio
[
]C. A median
[
]D. A Pearson's r
If you examine your output for Assignment 3, you will see only the F-ratio on your comparison of means test of these four choices. The F-ratio is a widely used inference statistic to compare means across groups. The median is a univariate measure. Pearson's r is a bivariate linear correlation coefficient and Chi-square is a different inference pdf often used with cross-tabulation tables.
When you examine a bivariate frequency distribution for gender and educational level in a sample with several thousand cases, you notice that the probability level or "P" = .0000. You decide:
[ ]A. The correlation
coefficient in the population is .0000
[
]B. You can't tell anything about the probability level
[ ]C. Your results
have a zero probability of occurrence
[
]D. Your results would happen by chance in less than one in 10,000 samples
D. Remember that nearly all the programs truncate the decimal places. In large samples, you often get a row of zeros when your results are highly statistically significant (sometimes you even get a 1.00 for the probability level instead).
BRIEFLY describe ONE way that computers can assist the researcher in analyzing data.
(There are several so I will leave it up to you to generate ONE response.)
-
level.
Often used to refer to Type One Error, typically
called the "statistical significance level" of
a finding. Alpha levels are set IN ADVANCE of data analysis. For example,
you decide ahead of time that you will set the critical region opf the
probability density function for rejecting the null hypothesis at .05 or
less.
2
Chi-square.
A
probability
density function often used to test whether
a sample association or correlation is nonzero in the population.
(it is NOT a correlation coefficient!)
Some widely used correlation
coefficients:
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Eta | Use with one independent variable (any level of measurement) and one numeric* dependent variable. Asymmetric. |
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Phi | Use with one nominal variable & one nominal (or ordinal) variable--symmetric. As a nominal measure, it can be used with any level of data (but it may not be the BEST measure for ordinal or numeric variables). |
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Rho | Use with two interval or ratio level variables--symmetric. |
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Tau-beta | Use with one ordinal & one ordinal (or numeric) variable--asymmetric. Can also be used with numeric variables but may not be the BEST choice in that case. |
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Gamma | Use with one ordinal independent & one ordinal dependent (or as an ordinal measure can also be used with numeric variables) variable (BAD inflated choice). Symmetric. |
*interval or ratio level variable = numeric.
REVIEW THE PROBABILITY CHART FOR READING PROBABILITY SYMBOLS BELOW:
We write the probability (p) of observing a relationship solely by chance as:
p =
or
p <
LESS THAN
or
p >
GREATER THAN
some figure between 0 and
1.
The probability of a sample
result occurring by chance in classical statistical inference is always
with reference to the null hypothesis IN THE POPULATION.
Probabilities
are always between 0 and 1. No exceptions to this rule here.
Here are some examples:
If there were NO relationship in the population (the correlation in the population is actually zero) then:
| p < .01
OR
"p is less than 1 chance in 100" |
the results in our sample would occur by chance less than once in 100 samples |
| p = .10
OR
"p equals 1 chance in 10" |
the results in our sample would occur by chance in 10 out of 100 samples |
| p > .05
OR
"p is greater than 5 chances in 100" |
the results in our sample would occur by chance in more than 5 in 100 samples) |
Although I would like everyone
to refamiliarize themselves with the < (less than) and > (greater than)
signs, remember that it is OK to write LT and GT on Exam 2 if you tend
to mix them up (e.g., p LT .001). And I know there are a couple of folks
who do.
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Here is EXHIBIT A. Cell entries are frequencies. You will need a simple calculator to work with some parts of Exhibit A.
This exercise is designed to give you practice with the parts of a table, cumulative percentages, and bivariate percentages.
Use these data to answer questions 1 - 6:
EDUCATION BY REGION OF ORIGIN (SOURCE: IMIU--I Made It Up)
REGION OF ORIGIN  |
EAST | SOUTH | MIDWEST | WEST |
| EDUCATIONAL LEVEL | ||||
| 0-4 YEARS |
5
|
10
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10
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5
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| 5-8 YEARS |
10
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5
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13
|
10
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| 9-12 YEARS | 25 |
15
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5
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20
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| 13-16 YEARS |
10
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10
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12
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15
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1. Which variable is the INDEPENDENT VARIABLE? BRIEFLY, how do you know?
REGION OF ORIGIN is certainly going to preceed educational level (or anything else) in time.
Without something of an explanation you may be a bit vague about HOW region might influence educational level. However, we can tentatively label this relationship asymmetric. We may not know exactly how region may affect educational achievement but we certainly know that education will not influence region of origin. Region OF ORIGIN came first in time.
Where you see questions like this, be sure to examine the names of the variables closely. Do they give hints about time order, as region "of origin" does? Might the variable name refer to a general disposition (e.g., "IQ score") as opposed to a specific attribute ("math ability")? Be alert to variable names and the distinctions among them in your readings.
2. What is the marginal for those with 9-12 years of education?
Add across the 9-12 YEARS OF EDUCATION ROW to obtain the ROW TOTAL:
25 + 15 + 5 + 20 = 65 is the marginal for this value of education.
3. What is the cell entry for Westerners with the greatest amount of education?
The greatest amount of education is the "13-16 YEARS OF EDUCATION" group. There are 15 Westerners with this educational level in the sample.
So the cell entry, i.e., the frequency in that cell = 15.
4. Which region has the lowest average educational level? Show how you know.
Level of education is ordinal as coded here, so the "average" will be the median category.
There are: 50 Easterners, 40 Southerners, 40 Midwesterners, and 50 Westerners.
Their medians are: Easterners 9-12 years; Southerners 9-12; Midwesterners 5-8 years, and Westerners 9-12 years.
Midwesterners have the lowest average EDUCATION.
I used the cumulative percent for each regional group separately to find the educational category where the cumulative percent jumped over 50 percent. That is the median category for each regional group.
5. What percent of Easterners have at most 8 years of school?
This is the same as asking "8 years or less."
So, look at the column that has Easterners ONLY. 5 Easterners have 0-4 years of school and 10 Easterners have 5-8 years.
5 + 10 = 15 Easterners with at most 8 years of school.
(15 ÷ 50) X 100 = 30% of Easterners have at most 8 years of school.
6. What percent of those with 0-4 years of school are Midwesterners?
Looking ONLY at the 0-4 years of education row, there are 30 people with 0-4 years of education.
Reading across the row, we have 5 + 10 + 10 + 5 = 30 of them.
10 of these 30 people with 0-4 years of school are Midwestern. To obtain the percent, then:
(10 ÷ 30) X 100 = 33.3% of those with 0-4 years of school are Midwestern.
For each of the following
variables, please indicate (1) whether the variable is nominal,
ordinal, interval, or ratio and
(2) IN ONLY ONE SHORT SENTENCE
describe
the reason behind your decision:
| 1. GRE score (Graduate Record Exam score) | 2. Age group: 18-23, 24-30, 31-40, 41-50, 51-60, 61+ |
| 3. Marital status (Married, Widowed, Divorced, etc.) | 4. Number of books in Strozier Library |
GRE, like most standardized tests, is constructed to be INTERVAL-level.
Age group is collapsed into six categories of unequal width (the highest category is open-ended and probably has a large category frequency) This variable is ORDINAL.
Marital status categories are simply names or tags. We can't rank them. This variable is NOMINAL.
Number of books
is a count variable with equal intervals and a fixed zero. This variable
is RATIO.
1. Was the statistic used
an appropriate one for that kind of data? Was it valid to use?
2. If not, what could have
been used instead?
3. Were the results "statistically
significant" (that is: REAL)?
4. How strong was the measure
of association (or wasn't it given at all)?
A. The investigator does
a difference of means test where the independent variable is ethnic group
(Black-White) and the dependent variable is the number of cigarettes smoked
per day. The t-value was t 2000 = -6.95, p < .001.
F3,849 = 320.75
p < .01 eta = .10
X2 (28)
= 8767.10 p < .001
= .26
NOTE: "levels of treatment" is a term often used with experimental designs to designate the different experimental groups. Despite the title, experimental interventions could be ANY level of data, nominal, ordinal, interval or ratio. To play it safe, consider this nominal data since we don't know what the interventions were, only that there were three different treatment groups.
On the other hand, pounds
of weight is clearly ratio (you can't weigh less than zero pounds.)
X2 (3) = 3.23 p > .05 tau-b = .32
FOR SUGGESTED
ANSWERS SEE THE TABLE BELOW:
| QUESTION | APPROPRIATE? | ALTERNATIVE | REAL? | STRENGTH |
| A | Yes (1 nominal, dependent ratio) | Appropriate used | Probably
p < .001 |
Not given, who knows? |
| B | INVALID (ordinal variables, can't do a mean) | Tau-beta | Who knows?
Invalid test |
Invalid test, who knows? |
| C | No (ratio dependent variable, nominal coefficient used) * | Best: Difference of means test | Probably
p < .001 |
Moderate (for coefficient
given, we don't know what eta would be) |
| D | Yes (both ordinal) Anyone who played the lottery at all played in more than someone who said "no" | Appropriate used | Probably NOT
p > .05 |
Zero because you cannot reject the Ho: Tau-b = 0. It doesn't matter what the sample tau looks like (e.g., 0.32), it's really zero. |
* Although the test used is not the most appropriate, it is a valid test for these data, and issues of statistical significance and strength can still be assessed.
For each of the following variable pairs,
| 1. Age in years and Income in dollars | 2. Political orientation* and attitude toward President Bush |
| 3. Race and region (e.g., East) of origin | 4. Race and dollars of earned income |
*NOTE:Political orientation is the General Social Survey "polviews" item, which runs from 1=extremely liberal to 7=extremely conservative. We can distinguish this from "party identification," for example, whether an individual considers themselves a Democrat, Republican, Socialist, Green Party, etc. Political orientation is typically considered ordinal, whereas party identification is often seen as nominal.
ANSWERS HERE:
1. Chronological age will preceed income in time. Your income may make you LOOK younger (trips to the spa) but can't make you BE younger. Because age and income are both ratio, use Pearson's r ("rho'). r is the only numeric coefficient available to use. It typically will be more informative than an ordinal coefficient coefficent.
2. It is likely that political orientation (which is a GENERAL predisposition) is a cause of attitude toward President Bush (attitude toward President Bush is a more SPECIFIC attitude) for MOST PEOPLE. With two ordinal variables and a designated independent variable (political orientation), Tau-beta is a good choice. If you aren't comfortable designatiing an independent variable in this case, use Tau-c.
3. Race and region of origin are both basically fixed at birth. Further, to think of either as somehow causing the other invokes the "giggle factor" so this is a symmetric relationship. Both nominal, no designated independent variable, use Phi.
4. Race clearly preceeds income in time. One nominal (independent) variable, one ratio dependent variable; you can do a difference of means test and use Eta.
|
|
Here is 'EXHIBIT B.' Cell entries are percentages. This problem is similar in many respects to Assignment 3. Use Exhibit B to answer questions 1 - 13.
THE RELATIONSHIP BETWEEN GENDER AND
GENERAL INTEREST IN SCIENCE
| GENDER | MALE | FEMALE |
| INTEREST IN SCIENCE | ||
| Very Interested |
52%
|
45%
|
| Moderately Interested |
41
|
45
|
| Not Interested |
7
|
10
|
|
100%
|
100%
|
|
| CASEBASES |
861
|
1014
|
Source: The NSF Surveys of
Public Attitudes Toward and Understanding of Science, 1999. N = 1875. MD
= 7
|
|
|
|
|
| Pearson Chi-Square |
|
|
|
|
|
|
| Phi |
|
| Tau-b |
|
| Gamma |
|
| Pearson's r |
|
| Eta |
|
1. Is gender nominal, ordinal, interval, or ratio?
You can't rank order the categories so it is NOMINAL.
2. Is the Interest in Science question nominal, ordinal, interval, or ratio?
Someone who is Very Interested in science is more interested than someone who is NOT. 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, Gender or Interest in Science? How do you know?
Trot out the (1) giggle factor, (2) time order, and (3) ease of change. Do you think your interest in science makes you male or female? Only in science fiction! Above are three reasons from the Guidelines why GENDER is the Independent Variable!
Review these guidelines HERE
4. Which of the correlation coefficients above is the BEST or the MOST APPROPRIATE to use to assess the relationship between Gender and Interest in Science?
Phi
5. What was the basis for your decision?
Gender is nominal and independent. Interest in Science is ordinal. Phi has to be our choice here (we can't take a mean on an ordinal variable.)
6. Is the gender difference on the Interest in Science question (CHECK ONE:)
[ X ] REAL or [ ] ACCIDENTAL?
7. What was the basis for your decision?
The probability
level for the Chi-Square was .00379. That means that if Chi-Square were
really zero in the population and there was no association between gender
and the science interest question, you would obtain sample results such
as these in only about 3.8 in 1000 samples. That is a VERY rare event.
So we reject the null hypothesis that X2 = 0 and conclude that
there is a nonzero relationship between gender and science interest in
the population.
|
|
This could be one of those wierd 3.8 in 1000 samples. That's why we put the probability level---so that we know just what our chances of being wrong are.
8. Was the correlation coefficient that you selected "statistically significant"?
We said the correlation was "real" in question 6. That means it is "statistically significant."
9. What was the probability level (the level of "significance") for this correlation?
p = .00379
you could also just say: p < .01
10. Was the relationship between Gender and Interest in Science linear, monotonic, nonlinear, or can't you tell?
We can't tell.
First of all,
gender has no rank order to the categories.
Second, there
are only TWO categories of gender so we can't tell anything about linear,
monotonic or nonlinear trends.
We need three categories in our column or independent variable to assess linearity or monotonicity. Plus both the independent and dependent variables must be at least ordinal.
11. What was the numeric value of the correlation coefficient between gender and interest in science that you selected?
Phi = 0.08
12. How strong was this correlation coefficient?
Very weak
13. What do you conclude about the association between gender and interest in science?
Gender has a real,
non-zero, but very small, very weak effect on how interested the person
is in science.
 |
READINGS AND ASSIGNMENTS |
OVERVIEW |
|
Susan Carol Losh October
26 2004
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