| Smokers | Non-smokers | |
| American males | 51 | 49 |
| American females | 34 | 66 |
A is positively correlated with B if and only if the percentage of As among Bs is greater than the percentage of As among non-Bs.
A is negatively correlated with B if and only if the percentage of As among Bs is less than the percentage of As among non-Bs.
A is not correlated with B is the percentage of As among Bs is the same as the percentage of As among non-Bs.
Here is the incidence of symptom and disease for 100 patients.
| Disease | No disease | |
| Symptom | 37 | 33 |
| No symptom | 17 | 13 |
There is no correlation here though 85% of the nurses thought there was a positive correlation between the symptom and the disease. The present/present cell was the best predictor of the subject's judgments; a high figure in that cell prompted a positive judgment.
Notice that for both the symptom group and the non-symptom group about as many have the disease as don't have the disease (slightly more have it than don't have it for both groups; 37-33 with symptom, 17-13 without symptom). Whether you have the disease or not, about twice as many have the symptom as don't have it.
Subjects are inclined to look only at select cells for pertinent information.
Another example: Does God answer prayers? Many say yes because many time prayers were successful. But what about the other cells?
Another example:
Subjects were asked whether Mr. Maxwell, a fictional person they were asked to imagine that they met at a party, was a professor. They were told he was either a professor or an executive, and that he belonged to the Bear's Club. Subjects were then asked what additional information they would like to have to make their judgment. For example, what percentage of professors at the party are members of the Bear Club, or what percentage of executives at the party were members of the Bear Club? 89% of the subjects wanted the first piece of information, but only 54% wanted the second piece, even though both pieces are relevant. (Also relevant is the information regarding the percentage of professors at the party.)
Studies have shown these tests to be useless as indicators of personality traits. But in studies in which pictures and trait-labels are associated in ways that reflect no correlations, untrained subjects still claim to "discover" that certain traits are correlated with certain aspects of the drawings. Even professionals maintain confidence in them after learning of their inefficacy. Similar results apply to Rorschach tests. Quote: "I know paranoids don't seem to draw big eyes in the research lab, but they do in my office." (Chapman and Chapman, 1967, 1969)
Some causal conditions are necessary conditions: the presence of oxygen is a necessary condition for combustion; in the absence of oxygen there is no combustion. "Cause" is often used in this sense when the elimination of the cause is sought to eliminate the effect (what's causing the pain?)
Some causal conditions are sufficient conditions: the presence of a sufficient condition the effect must occur (being in temperature range R in the presence of oxygen is sufficient for combustion of many substances. "Cause" is often used in this sense when we seek to produce the effect (What causes this metal to be so strong?)
Looking for special circumstances: what was the cause of the fire? Oxygen? or an arsonist's match?
Causes are sometimes said to be INUS conditions in that they are Insufficient but Necessary parts of an Unnecessary but Sufficient set of conditions for the effect. Striking a match may be said to be a cause of its lighting. Suppose there is some set of conditions that is sufficient for a match's lighting. This might include the presence of oxygen, the appropriate chemicals in the matchhead and the striking. The striking can be said to be a necessary part of this set (though insufficient by itself) because without the striking among those other conditions the match would not have lit. But the set itself, though sufficient, is not necessary because other sets of conditions could have produced the lighting of the match.
2. Correlations are about actual populations and are not lawlike. Causal relationships are lawlike in the sense that they are about hypothetical populations as well as actual populations. When A is said to be the cause of B we are saying that were there an increase in the incidence of A there would be an increase in the incidence of B; or if A cases were to diminish, B cases would diminish, too. (If fewer people smoked, there would less lung cancer.) Mere correlations pertain only to actual populations. If National League success in the Super Bowl is merely correlated with stock market decline, then we should not expect changes in the stock market to affect the outcome of the Super Bowl (or vice versa).
How can one form judgments about causal relationships based on statements about correlations?
For example, there is a strong positive correlation between an increase in the number of sex education classes and an increase in the rate of gonorrhea. Suppose we conclude that increasing the number of sex education classes has caused the increase in the gonorrhea rate.
(A) Is the statistical premise (the statement about the correlation) true or well founded?
(B) What alternative explanations are available?
1. The correlation might be accidental or coincidental. Increase in the national debt is positively correlated with an increase in the gonorrhea rate, but there is no causal connection.
2. The relation might be spurious, both an increase in the number of sex education classes and an increase in the rate of gonorrhea being the effects of the same cause.
3. The causal direction might be the reverse. Could the increase in the gonorrhea rate be causally responsible for the perceived need for more sex education classes?
4. The causal relation might have been more complex than the conclusion suggests. The increase in sex education classes might have caused a change in attitudes about sex, which led to an increase in sexual activity, which led to an increase in the gonorrhea rate.
5. The causal relation cited might be insignificant relative to other factors responsible for the increase in the gonorrhea rate.
There is a strong positive correlation between the number of fire trucks in a borough of NYC and the number of fires that occur there.
There is a strong positive correlation between foot size and hand writing quality.
There is a strong negative correlation between the number of forward passes thrown in a football game and winning the game.
Heavy coffee consumption is positively correlated with heart attacks.
Going to the hospital is positively correlated with dying.
An increase in the number of hours kids watch TV positively correlates with decrease in SAT scores.
Marijuana use is negatively correlated with high GPAs.
Another example:
"[W]hile half the country's communities have flouridated water supplies and half do not, ninety percent of AIDS cases are coming from flouridated areas and only ten percent are coming from nonflouridated areas."
Any connection?
1. Communities aren't all the same size: flouridated communities (likely to be big cites) might contain much more than half the population.
2. The relationship might be spurious: cosmopolitan/progressive attitudes might encourage both fluoridation and lifestyles associated with AIDS
Another example:
Is there a causal relationship between class attendance and grades achieved?
"Students with the lowest attendance earned the poorest grades. Those who attended 79 percent of the classes or less ended up in the low C range; 90 percent and above scored above a B average. Student who sat up front got 'significantly higher grades,' but Walsh [the researcher] thinks they could be more interested in the subjects."
A is not a necessary condition for B if B occurs without A.
Which factor is always present when the effect is present?
If among the residents of a dormitory there is a rash of stomach upsets, we would likely look for one food item that all the patients ate as the cause.
1. The conclusion applies only to the occurrences considered.
2. Only probable: other important conditions might have been overlooked; it might have been a combination of factors
Which factor is always absent when the occurrences of the effect are absent?
Five factory workers are found to be inefficient relative to others who are doing the same work. The efficient workers and the inefficient workers were found to be similar in all relevant ways except one: the inefficient were not part of a profit sharing plan. Conclusion: profit sharing causes efficiency.
1. The conclusion applies only to the occurrences considered.
2. Only probable: other important conditions might have been overlooked; it might have been a combination of factors
Which factor is always present when the effect is present?
Which factor is always absent when the occurrences of the effect are absent?
Eight patients have a disease and each was given some remedy or other. Four patients who are given serum S are cured. Of those who are cured no other single remedy was given to all. Of the four who were not cured, every patient was given at least one of the remedies (but none the serum S). Serum S judged to be the cure.
1. The conclusion applies only to the occurrences considered.
2. Only probable: other important conditions might have been overlooked; it might have been a combination of factors
The factor is the only one that is present when phenomenon is present and absent when the phenomenon is absent.
Two identical white mice in a controlled experiment were given identical amounts of four different foods. In addition, one of the mice was fed a certain drug. A short time later the mouse that was fed the drug became nervous and agitated. The researchers concluded that the drug caused the nervousness.
1. Less general conclusion than the inverse method of difference, which applies to all occurrences listed
Use the direct method of agreement to isolate necessary conditions (if no factor, no effect) and the method of difference to isolate those that are also sufficient.
1. Less general conclusion than the double method of agreement, which applies to all occurrence listed;
George, who exercised regularly, took vitamins, and got plenty of rest, contracted a rare disease. Doctors administered an antibiotic and the disease cleared up. convinced that the cure was caused by either the exercise, the rest, or the antibiotic, the doctors searched for analogous cases. Of the two that were found, one got no exercise, took no vitamins, and got little rest. He was given the same antibiotic and was cured. The other person, who did the same things George did, was given no antibiotic and was not cured. The doctors concluded that George was cured by the antibiotic.