Association Flashcards
What is association?
Association is a property of two or more variables. In this example, the two variables are the amount an individual spends on healthcare, and the number of additional years the individual survives. Association is not the same as causation: Two variables can be strongly associated but have no causal connection, or can have a causal connection and no discernible association. Concluding that association implies a causal relationship is the post hoc ergo propter hoc fallacy.
What is the correlation coefficient r?
The correlation coefficient describes the agree of association between two variables and has a value between -1 and 1. The correlation coefficient, r, is positive when variables are positively associated; the value of r is negative when variables are negatively associated.
Sometimes we shall use subscripts to clarify which correlation coefficient we are talking about: The symbol rxy denotes the correlation coefficient for X and Y. The correlation coefficient for a scatterplot of Y versus X is always equal to the correlation coefficient for a scatterplot of X versus Y symbol (rxy=ryx)
What is positive and negative association?
Positive: When one variable increases so does the other
(Correlation coefficient larger than zero, r>0)
Negative: When one variable decreases so does the other
(Correlation coefficient less than zero, r<0)
What is a secular trend?
A linear association (trend) with time.
What is the correlation coefficient not good at describing?
The correlation coefficient r measures only linear association: how nearly the data fall on a straight line. It is not a good summary of association if the scatterplot has a nonlinear (curved) pattern.
The correlation coefficient r is not a good summary of association if the data are heteroscedastic.
The correlation coefficient r is not a good summary of association if the data have outliers.
What is the five-number summary of football-shaped scatterplots? (Football-shaped = linear, homoscedasticity, no outliers)
Football-shaped scatterplots can be summarized rather well by five numbers:
the mean of X, the mean of Y, the SD of X, the SD of Y, and the correlation coefficient r.
What is the ecological correlation?
Ecological correlation is the correlation coefficient calculated for averages of individuals, rather than for individuals. Ecological correlations say little about the (linear) association for individuals; generally, ecological correlations tend to overstate the strength of the association for individuals.
Correlations of averages of measurements can differ enormously from correlations of individual measurements.
Typically, they are much larger, but they can be smaller, too.
What are standard units and how do you convert data to standard units?
Standard units are a way of putting different kinds of observations on the same scale. The idea is to replace a datum by the number of standard deviations it is above the mean of the data. If a datum is above the mean, its value in standard units is positive; if it is below the mean, its value in standard units is negative. A datum that is above the mean by 2.5 times the SD is 2.5 in standard units.
datum in standard units = (original datum - mean of data)/(standard deviation of data)
What does the sign of a value in standard units imply?
Values that are larger than the mean are positive in standard units
Values that are less than the mean are negative in standard units
How is the correlation coefficient, r, calculated?
The correlation coefficient r of two variables X and Y is the average of the product of X in standard units and Y in standard units. You must be sure to multiply the measurements corresponding to the same individual.
How is the correlation coefficient, r, affected by transformations?
The correlation coefficient is unaffected by affine transformations of both variables if the multiplicative constants in the transformations are both greater than zero or are both less than zero If one of the multiplicative constants is less than zero and the other is greater than zero, r changes sign.
