Guess what!  As the weeks have been passing, and you've been teaching 
us all this research data stuff...  I've asked myself a hundred times - 
"when the hell am I ever going to need to know all of this?"

The past few days I was in Chicago with J'ona Meyer for a CJ 
convention and during one of the presentations a gentleman put up his data on a 
projector......and I was ASTONISHED that I understood EVERYTHING he was 
talking about - from his variables, to the standard deviation, to it's statistical
signifigance, etc.!  I'm happy to report I'm actually learning 
something of use in your class!  Had I not taken this class, I would have sat there 
feeling like a total dummy not knowing a thing what he was talking about.

Thanks - Denise Gilboy

Some concepts:

Rate:  A statistic that reduces numbers to a common base.  The base is often, but not necessarily, the total population in an area.  If we are looking at voting participation, we might compute rates using the base of the number of adults 18 or over.  If we are trying to predict an election, we might use a base of registered voters. 

A crude birth rate is the number of births per 1,000 population.  Fertility rate is the number of births per female during her lifetime. 

Time Series analysis:  uses time periods as the unit of analysis, looks at how things change over time often in one case.   A lagged time series takes into account the time it takes for one variable to influence another, thus incarcerations in one year might be related to crimes in the next year.

 Cross-sectional analysis compares a number of cases at one point in time.

Reliability:  are statistics computed the same way in different geographic units or different time periods.  This causes all sorts of problems - it is better to imporve statistics, but doing so causes us to lose comparability. 

Validity:  do the statistics measure what we want them to measure.  Crimes reported to the policy are not a valid measure of the amount of actual crime, especially for crimes that are often not reported. 

Case oriented vs. variable oriented.  The case oriented approach is more qualitative, although quantitative trend data can be used.  The variable oriented approach assumes that the same variables are causally related in the same way in a large number of cases, e.g., "capital punishment" and "homicide rates" in a number of states or countries. 

Outliers:  especially in variable-oriented research, it is important to look for exceptional cases that are very different from the norm.  These tend to cause a disproportionate impact on our results. 

Lagged:  Using statistics from past years to predict events in current years.  This is done because our theory says that causal linkages take some time to take place.

For the exam, you should know how to set up the regression equations to fit a path diagram. All you have to do to actually do a path analysis is put these equations into Microcase. The rules are the follows:

1. There should be a regression equation for each variable that has an arrow pointing towards it.
2. For each equation, the variable having arrows pointing into it is the dependent variable, and goes to the left of the euqals sign.
3. For each equation, the variables on the left of the dependent variable that have arrows pointing into it are the independent variables.  These are listed to the right of the equal sign and connected with + signs.
4. There is no need to include an intercept, because we are interested only in the standardized regression equations or beta weights.

    alienation from government = status deficiency

Today we will look at testing causal hypotheses.  On page 93 in the text, we have the example of the relationship between Height and Liking Basketball.  This is anIV and a DV.  An obvious TEST VARIABLE is Gender.  This would be Antecedent, Gender determines both your height and liking for basketball.  We could draw this as a path diagram (on board).

When we introduce the control, we split the table into two parts, e.g.,

                             Males                  Females                 Total
                        Tall     Short           Tall   Short         Tall    Short

Likes BB           85%    85%            25%   25%         65%    45%
Does Not           15%    15%            75%   75%         35%    55%

Total                 100%  100%          100%  100%      100%   100%

In the real world, things are never this sharp.

Let's look at some real data, using FEAR WALK, PLACE SIZE and R.INCOME from the GSS data set:

In the total sample, the low income respondents are more likely to feel there are areas near them where they should fear walking.  However, this effect disappears for some of the respondents when we control for the size of the town in which they live.

To make it a finished Table:
                Small Town or rural         Small City        City/Surb          Total
                  Low  Med  Hi             Low Med Hi        Low Med Hi      Low Med Hi

Fear Walk         30%  27%  24%            48  42%  20%      56  41  43     51%  39%  41%
No Fear           70%  73%  76%            52% 58%  80%      44% 59% 57%    49%  61%  59%
                     p = .710                p = .043           p = .000       p=.000
                     N = 251                 N = 133            N = 1253      N = 1637
 

To to a more complete causal model of Fear of Walking at Night, we should introduce more variables.  Some of them may be in our data set, others now.  

What variables should we look at?

Variables      Hypotheses
Gender           Females more fearful than males.
Age              Elderly more fearful, also Children.  Might be curvilinear.
Crime Rate       People in high crime communities
Street Lighting
Freq of Patrols
Graffiti, Broken Windows, Trash, other indicators of an "out of control" neighborhood
Bicycles
Number of Pedestrians
Physical Shape
Training in Self Defense

We can examine some of these variables with our data.  We may find it useful to use regression rather than cross-tabulation.
We can also use pages 114-122 in the workbook as examples..

The Art and Science of Cause and Effect. (powerpoint)

Probabilistic cause, not an absolute cause, not a cause that is sufficient or necessary.   "Cigarette smoking causes
cancer."  WHat we mean is, smoking cigarettes increases the likelihood of getting cancer.  How much?

There are multiple causes for everything.  What we want to find out is how much each thing contributes.  There are also
causal linkages, or indirect causes.  A causes B and then B causes C.

Diagraming causal models.  We put the dependent variable at the right.  We draw arrows going into it for each causal
variable that effects it directly.  Then we can have arrows that go into the arrows, steps into the causal analysis, as in
this sample file:
http://crab.rutgers.edu/~goertzel/homomale.htm

Criteria of Causation - how do we know that something is a cause of something else.

1.  Time Order.  The cause comes before the effect.  Sometimes we sort out the time order theoretically, we assume that
education preceeds employment.  Or we can use a research design that involves gathering data at two points in time.  If
you don't have measurements at two points in time, this is shaky.

2.  Correlation.  The two variables vary together.  When one is high, the other is high OR when one is low the other is
high.  This gets at the degree of causation, the higher the correlation the strong the causal relationship.

3.  non-spuriousness,  we want to know that the correlation is not cause by something else.  We can test this with an
experimental design, if feasible.  Or we can use statistical controls, which are not quite as convincing but its all you do
in many cases.

We test for non-spuriousness by introducing controls.

Causal Models:  representations of the complex causal relationships between variables.  Variables have different causal roles, but this is determined by our causal our causal model, it is not inherent in the variables.   One person's cause can be another's effect.

Dependent Variable - that is what we want to explain.  Often these are opinions or behaviors

Independent Variable - what we use to explain it.  Often there are traits or physical characteristics, e.g., sex or race,
almost always independent.

If you study the relationship of race on voting, for example, race would be independent and voting dependent.

Antecedent variables, things come before the independent variable.  This helps us to deal with a causal chain.
Antecedent variable cause IV which causes the DV.
If the antecedent variable "explains" the relationship, we have an "explanation", we say it is "spurious".

Intervening Variables, this that are intervening, e.g.   Race determines ideology which determines the vote.
This is an "interpretation" it tells WHY the causal relationship exists.
Path Models:  a way of graphically expressing complex causal models.

Example:  Determinants of Adult Homosexuality in White Males.

Example:  The Seattle Social Development Project. 

  Today we will learn the formula for margins of error for mean scores:

If you need a margin of error for a mean score (an average such as income in dollars or scores on a test), you need to know the standard deviation (sd) and the sample size (N). Ignore any other information you are given, including the size of the population.
Use the following formula: M = 2 * sd / SQRT(N).

Here is an example question:  A study of Rutgers Camden Sociology Department graduates showed that the mean annual salary was $55,000 with a standard deviation of $3500. Three hundred graduates were sampled. What is the margin of error for this statistic?    Answer:  M =   2 * 3500/SQRT(300)  =  7000/17.3205  = $404.15.  Note that this is a dollar amount, since the question was in dollars.  It is not a percentage.

What is the confidence interval for this mean score?  The answer is $55,000 plus or minus $404.15, or $54,595.85  to $55,404.15
The formula for percentages or proportions is:

m = 1/sqrt(n)   

Take 300, the square root of 300 is = 17.32     1 /17.32 = .0577  * 100 = 5.8%

This formula is for  proportions or percents (if you move the decimal over two)
  m = 1/sqrt(n)    Solve for N:      m² =  1/n      n * m² = 1     n = 1/ m²    If we need a margin of error of 3%, or .03.   n = 1/ .03²

  If you have a sample size and need to know the margin of error, use    m = 1/sqrt(n)

   If you are given a margin of error and asked how large a sample you need, use  n = 1/ m²

          In these formulas n = the size of the sample (not the population).    m = the margin of error expressed as a proportion, not as a percent.  Thus, if the questions says "we need a margin of error of 5%, then m = .05.   

If our sample is stratified, this means we really have several sub-samples and we need the same size sample for each of them, regardless of the size.  For example, if we want sample white, black and Hispanic respondents and make statements about each group, we need the same size sample of both regardless of their size in the population.  Thus, if we need a margin of error of 5% for each of the three groups, then the answer is  3 * ( n = 1/ m² ).

If you need a margin of error for a mean score (an average such as income in dollars or scores on a test), you need to know the standard deviation (sd) and the sample size (N). Ignore any other information you are given, including the size of the population.
Use the following formula: M = 2 * sd / SQRT(N)

Terms:

Margin of Error:  How much a sample statistic is likely to vary from the population parameter.  We say that we are 95% sure that the sample is not off by more than the margin of error.  How this is presented in NY Times.  "19 out of 20" is another way of saying 95%. 

 Confidence level:  we always use a 95% confidence level.

1. we may not have a list of the population.  If we do not, we first divide the sample into sub-groups of some kind (census tracts, blocks, classrooms, organizations, depending on the nature of the study).  We then sample the subgroups and list the populations in them .  This is called cluster sampling
2. We may be interested in differences between sub-groups of the sample and need to make sure we have enough of them.  In this case we select random samples of each of the relevant sub-groups, and weight the results appropriately.   This is called stratified sampling. 
   
3. Sometimes we just go down a list, which is called systematic sampling.  This gives the same results as simple random sampling, unless there is some systematic ordering to the list that causes a distortion
4. Sometimes we use non-random or "quota" sampling.  This is done for convenience, or because we just want to know what the range of differences is without putting numbers on them.
   

                  An example:  a study of UFO Abduction Status.

Dichotomous Measurement -   Two and only two categories.  Can be a natural dichotomy or a  "dummy variables" - we take a complex variable and divide it into a series of dichotomous variables.  

Nominal Measurement.  Categories that could be put in any order.
      Catholic, Protestant, Jewish, Moslem, LDS, Buddhist, Episcopalian, Baptist
                       variable one, category of religion, variable two denomination.
            Mental illnesses (DSMIV) e.g.,  adjustment disorder, borderline personality disorder, paranoid schizophrenic
               Crimes:   burglary, assault, murder.  What do these terms mean?  Look at the US Criminal Code. 

  Each individual should go into one and only one category on a variable, one value on a variable.   For example:  What is your favorite food, we have a long list, but each person is allowed only one.
       Sorting people into categories must be reliable and accurate or valid.

Ordinal Measurement.   Here we have categories in a logical order.       Very short, short, medium, very tall, tall .  Often we take continuous variables and make them ordinal.    Income:   Under $20,000   $20 to 40,000  $40 to 60,000   $60000 plus.

Interval Measurement:   TEMPERATURE IN FAHRENHEIT OR CENTIGRADE, 0 degrees is not the absence of heat.  How about the day that the "temperature doubled" in New York City?

Ratio Measurement:    Income in dollars:  a continous numerical value PLUS a meaningful zero point.  Height in inches. 
 
Scaling is when we use a number of measures, such as test scores or questionnaire items, to measure a more general concept.  This often allows us to move to a higher level of measurement.  For example, we can add up test score items them up (in which case your text would call it an "index", although many people still use the form scale) , or they may be ordered from lowest to highest (in which case it is a true scale as the term is used in your book).  Your test is an example.  I just add up the points, to measure the general variable "knowledge of research methods as covered in the first part of the course."  Another approach would be to rank the items from easy to hard and see which you could do.  This is tricky, because some people can do the hard ones and not the easy ones.  When we make an index or scale, we get measures that can be treated as interval, even if they are not strictly interval.  Scaling methods can be more precise, but these are not used much in sociology or CJ.  For example, we could scale the seriousness of crimes.  There are various methods of measuring this. - paired comparisons means asking a sample of people to rate crimes based on their perceived seriousness.

One of the reasons we have to be clear about levels of measurement is that the statisitcs we use depend on how the data are measured. 

Statistics for Nominal Data:    Percentages and Chi Square   The percentages are descriptive (they summarize our data), the chi square is inferential (it tells us if we can generalize from our sample).   Survey data usually produces nominal (or ordinal) statistics.   Cramer's V is a correlation coefficient for nominal data, scores on it vary from 0 to 1, but there are no negatives since the data are not ordered.

Statistics for Ordinal Data:  The median is the only statistic we have covered that is specifically designed for ordinal data - it finds the case in the middle once all the cases are sorted in order.  There are correlation coefficients for ordinal data which you can find on the "statistics" page for crosstabulations (gamma, tau) but it is more common to use interval statistics (Pearson's r) or nominal ones (Cramer's V) with ordinal data.

Statistics for Interval Data:   Scattergrams, means, standard deviations, correlation coefficients.  Tests of statistical significance for correlations.

Today we will begin with Amar Patel's Chi-Square lesson.   This covers the concept of expected frequencies and observed frequencies, and introduces the concept of "fairness", the difference statistic and the chisquare statistic.  These are applied to problems where the expected frequencies are given by a null hypothesis of "fairness".

We can apply this to any distribution where we have a theoretical reason to expect a certain result.  E.g., with two dice, each with six sides.  What results are possible and what likelihood do we have?

1.  
   
2.   *               Snake-eyes!
   
3.   **             (1 and 2; 2 and 1)
   
4.   ***           (1 and 3; 3 and 1; 2 and 2)
   
5.   ****         (1 and 4; 4 and 1; 3 and 2; 2 and 3)
   
6.   *****       (1 and 5;  5 and 1; 4 and 2;  2 and 4; 3 and 3)
   
7.   ******      (4 and 3;  3 and 4;  5 and 2;  2 and 5; 6 and 1; 1 and 6)
   
8.   *****       (4 and 4; 5 and 3; 3 and 5; 6 and 2; 2 and 6)
   
9.   ****         (5 and 4;  4 and 5; 6 and 3;  3 and 6)
   
10.   ***            (5 and 5; 6 and 4; 4 and 6)
    
11.   **              (6 and 5; 5 and 6)
    
12.  *                  Boxcars!

25 men agreed
17 men disagreed
65 women agreed
30 women disagreed
 
 

Observed Frequencies or Obtained Frequencies	Men	Women	total
Agree	25	65	90
disagree	17	30	47
total	42	95	137

    We can compute expected frequencies, based on the null hypothesis that men and women do not differ intheir opinions.  We can compute these knowing only the marginal or total frequencies.  The easy way to compute them is to multiple the row total for each cell by the column total for that cell, then divide by the grand total.  Another way would be to convert the row totals to proportions, then multiply then by the column totals.  Expected Frequencies - rt *ct /gt

Expected Frequencies	men	women	total
agree	90*42/137=27.59	90*95/137=62.41	90
disagree	47*42/137=14.41	47*95/137=32.59	47
total	42	95	137