RMHI 3. Contingency Tables

unaffected, coverage

Statistical assumptions can differentially affect estimators, in that the sample statistic may be reasonably _______ by some violations (e.g., normality) but the _______ of the confidence interval for that sample statistic may be dramatically affected.

df

The χ2 distribution has only one parameter, its ___. The number of ___ determine the shape of the distribution, as shown in the above graph of several members of the χ2 family.

df

The χ2 probability distribution is a family of related-distributions defined by the same mathematical function. Different "members" of this family arise from different values for a ___ parameter that is part of the function (the concept of __is explained in the next subsection).

observed, expected

Compare the ______ cell frequencies foij to the _______ cell frequencies feij to see how much the former deviates from the latter (where i & j indicate a category in 1st & 2nd variables respectively).

unbiased

Confidence interval coverage of the three OR is ______ because the mean coverage for each sampling distribution is reasonably close to 95%, irrespective of sample size (if we had used, e.g., 10 times as many replications then the coverage would be extremely close to 95%).

P/(1-P)

= the odds of an event occurring

consistent

A biased estimator may still be ________ if the mean of the sampling distribution gets closers to the population parameter value as sample size becomes larger.

stronger, larger

A greater degree of dependency reflects a _____ association between the two categorical variables, which in turn will correspond to a _____ observed χ2 test statistic value.

deviation

A stronger association is indicated by an estimated value having a larger ______ from 1 (i.e., by a value either closer to 0 or further towards +∞).

efficient

All three estimators of OR are about equally _______, because the sampling distributions do not differ noticeably in shape and spread at each sample size.

biased

All three estimators of OR value are ______, because the means of the sampling distribu- tions do not equal population OR value of 5.5 (this is especially true for small n).

non-negative

All χ2 values are __-_____, because they are squared quantities by definition. The critical rejection region in a null hypothesis test for any χ2 distribution is in the upper tail only. Therefore, bigger observed χ2 values correspond to smaller obtained p values.

smooths

An adjusted OR estimator that ______ cell frequencies

0.5

An adjusted OR estimator that adds ____ to each cell.

efficiency

An estimator has greater _________ if it displays smaller variability (i.e., a smaller standard error) compared to a second estimator, when sample size is the same.

unbiased

An estimator is ________ if the mean of the sample distribution equals the population parameter value, irrespective of sample size.

expected

Calculate a pattern of cell frequencies that would be ______ if there was no association (i.e., independence) between the two categorical variables.

equal

Expected frequencies always have the following properties: • The expected row marginal frequencies _____ the observed row marginal frequencies. • The expected column marginal frequencies _____ the observed column marginal frequencies. • The total sum of all expected cell frequencies _____ the total sum of all observed cell frequencies (which equal the total n).

biased, consistent, biased

If both variables are not normally distributed, then. . . • The shape of the sampling distribution can change quite dramatically (e.g., compare the graphs for n = 10 when data are normal or non-normal). • The point estimators for sample correlation are slightly more _______, but they both tend to be ________ (see the values in the boxes in the upper left corners of each graph). • However, confidence interval coverage becomes much more noticeably ______—the intervals liberal (i.e., the actual coverage is smaller than the nominal 95% coverage, as shown by the values in that same box). • Confidence interval coverage is also not consistent, because the degree of bias does not decrease as sample size increases.

unbiased or consistent

If the point estimator of a sample statistic is ______ or ______, it does not necessarily follow that the coverage of the corresponding interval estimator will also be ______ or ________.

positively, non-normal

In each instance, the sampling distribution of the odds ratio is noticeably skewed _________ and is therefore ___-_____

dependency

In general, a greater discrepancy between the observed cell frequencies and the expected cell frequencies indicates a greater degree of _______ between the two variables comprising the contingency table.

1

Odds ratios have the following properties: • If an OR = __, then the two variables in the table are independent and have no association.

precision

Note that interpreting a confidence interval in this way solely as a proxy null hypothesis test is a very limited use of the technique. The degree of __________ shown by the width of the interval, the closeness of either of the two bounds to one, etc, are also important features to use in making an overall inference.

0

Odds ratio considered undefined if any observed cell frequency is ____ (without further amendment—e.g., +0.5 to all cells).

negative

Odds ratios can never be ______—their range is 0 < OR < +∞.

<1, >1

One notable problem in interpreting an odds ratio (both a point estimate and a confidence interval) is that values that are ____are more difficult to understand than values that are ____. Intuition and everyday experience mean that we have a better inherent understanding of an odds ratio of e.g., 2.5, than we do when the odds ratio equals 0.4—it is easier to understand in the former that the odds are 2.5 times larger (it is not so immediately clear what the value 0.4 means).

frequency counts

Summary data of this kind are often more fully called ______ _______, with the typical display of such data being either in a frequency table or a bar graph.

higher

That is, we can state that the odds of choosing Coke are (10 × 11)/(5 × 4) = 5.5 times ______ for people in the Positive Coke Condition than for those in the Positive Pepsi condition.

columns, rows

The "+" used in the subscripting of this notation serves a specific purpose. It indicates whether the summing up of the joint cell frequencies fij has occurred either over the ______ or across the _____.

actual, 95

The _________ coverage of the confidence interval in the simulation is therefore the proportion of the 10,000 confidence intervals that captured the true population value. If the interval estimator is unbiased, then we would expect the actual coverage percentage to be close to the expected ____ percent value.

generalise

The above results for both normal and non-normal distributions are not definitive for all correlations under all conditions, as only two correlation values, four different sample sizes, and one kind of non-normality was investigated. These results may therefore may not necessarily _________ to any degree of non-normality or to larger sample sizes.

chi square

The calculation we have just gone through is often called a ____ ____ test for a contingency table, because our Tobs value of 4.821 has a theoretical χ2 probability distribution. That is, for the χ2 test: Tobs ~χ2(df). The χ2 test is so named because it uses the theoretical χ2 probability distribution as the basis for making inferences from the sample observations to the population level. • That is, if we constructed a transformed sampling distribution for Tobs when it is cal- culated from a contingency table, then the sampling distribution would correspond to a theoretical χ2 probability distribution. • The "χ" character is a lowercase letter from the Greek alphabet (upper case chi is like roman "X") (sadly, we all have to suffer the consequences of early statisticians being trained in the classics at Oxford and Cambridge).

sample space

The classic illustration is tossing a coin, or throwing dice. The set of all possible outcomes is sometimes referred to as a ____ _____, which may be either finite (i.e., countable) or infinite. Typically, the notation P or Pr is often used to indicate a generic probability value (and sometimes it is easier to express a probability in percentage terms—i.e., P × 100.

expected cell frequency

The general formula for calculating the ____ _____ _____ (signified by feij ) of a cell in a contingency table is the product of the row marginal frequency (signified by fi+) and the column marginal frequency (signified by f+j ) for that cell, which is then divided by the total sample size (signified by n). That is, feij =(fi+×f+j)/n

random variable

The latter type may seem less clear, at this stage of the lectures, than the former, because the idea of any sample statistic having different values from one sample to the next is probably starting to become clear (if not completely clear already). However, this same applies to the two bounds of a confidence intervals. Each bound is a _______ _______, and therefore the value of the lower and upper bound of a confidence interval will change from one sample to the next, in general.

normal

The main point to take from the last observation is that we should not assume that all sampling distributions will be _______—the shape of a sampling distribution varies according to the sample statistic used to form it (which is why we have different corresponding theoretical probability distributions).

same

The odds ratio does not depend on which variable defines the rows and which one defines the columns of the 2 × 2 contingency table—the _____ OR value is obtained.

efficient

The sampling distributions when data are non-normal are much less ______, especially for small n, than the corresponding ones obtained when the data are normal (i.e., the non-normal ones have larger standard errors).

association

The structure and form of such a two-way cross-classified table prompts asking whether the joint frequencies in the cells indicate that an ________ may exist between the categories of the two variables. That is, does the relative frequency of people in the categories of one variable covary in a systematic way with categories in the other variable?

CI

The utility of the odds ratio as an effect size measure is enhanced because a ___ can be calculated for it. This implies that an odds ratio is both: (i) A sample statistic when calculated on sample data; and (ii) An (unknown) population parameter defining the population strength of association. SPSS provides a 95% confidence interval for estimated odds ratios. • The 95% CI from SPSS in this instance is (1.15, 26.41). NB: The MATLAB program XECI (used in lab classes) provides more flexibility.

cross-classified, associated

These frequency counts can be represented as a ____-______ contingency table, whereby each cell in the table uniquely identifies the number of people who are jointly defined by pairs of categories from each variable comprising the table. By doing so, we can then investigate whether the frequency of people in categories of one variable are ______ with categories of a second and different variable.

reciprocal, lower

This difficulty can be alleviated by (i) taking the _______ of the estimated values, and then (ii) explaining the result in terms of the odds being _______ or reduced.

inverting

We can always shift from odds of winning to odds of loosing (or vice versa) by just _______ the value of the initial odds (i.e., odds of 19/1 to win is the same as odds of 1/(19/1) = 1/19 to lose).

1

When the 95% confidence interval for an odds ratio does not capture the value of ____ between the lower and upper bound values, we immediately know that the observed sample odds ratio value (e.g., our point estimate of 5.50) is significantly different from a null hypothesised value of 1 (i.e., from independence).

consistent

all three estimators of OR are ____ because the means of sampling distribution get increasingly closer to 5.5 as sample size gets larger.

odds

are the probability of some event occurring compared to the probability of that event not occurring, i.e. it is an expression of relative probability

cumulative percentage

as well as frequency counts for each category, the table also has the percentage of the sample in each category, and the ______ _______ of people over all categories.

(I-1)*(J-1)

df for chi square test

dependency, independent

does the relative frequency of people in the categories of one variable covary in a systematic way with categories in the other variable? if they do...then there is a _______ between the frequencies of one categorical variable and the frequencies in the other; i.e., there is a contingency between the two variables—which prompts calling it a contingency table. • If they don't . . . then the frequencies of the two variables are deemed to be ________.

odds ratio

effect size for a 2x2 contingency table

feij=(fi+*f+j)/n

feij=

sampling variability

how do we delineate dependency from independence? That is, does any observable contingency in cell frequencies just result from ______ _______ or does it represent a meaningful population association?.

odds ratio

indicates the odds of one category occurring in one variable relative to the odds of a second and different category occurring in another variable.

a*d/b*c

odds ratio formula

strength, precision

researchers tend to report an odds ratio and its 95% confidence interval as an effect size that indicates both the _____ (i.e., the sample odds ratio value) and the ______ (i.e., the two bounds of the confidence interval for the sample odds ratio) of the effect being observed in the sample data.

probability

the term refers a measure of uncertainty expressed as the long run relative frequency of one outcome occurring among the set of all possible outcomes that could occur over many replications of same process.

symmetry

there is a ______ in the way that the odds ratio can be interpreted in words.

sample size

variability in the sampling distributions of the three OR gets smaller with larger _____ ______ (i.e., standard errors are smaller)—remember that the standard error uses n in the denominator of its calculation.

point, interval

we will consider properties of bias and consistency for (i) _______ estimators, which is an alternative term used for the calculation of sample statistics, and (ii) ______ estimators, which is an alternative term used for the calculation of confidence ________.

x^2obs

what is the observed test statistic in a contingency NHST?

consistent, unbiased, unbiased

• The usual sample correlation coefficient is _______ (but it is slightly biased for small sample sizes) • However, the confidence interval coverage is ________. • The unbiased estimator for sample correlations and for confidence interval coverage are both _______ (as would be expected!).

lower

• the odds of choosing Coke are 5.5 times _____ for people in the Positive Pepsi Condition than for those in the Positive Coke Condition. • we are 95% confident that the odds of choosing Coke are between 1.15 and 26.41 times ______ for people in the Positive Pepsi Condition compared to people in the Positive Coke Condition.