Читать книгу Applied Biostatistics for the Health Sciences - Richard J. Rossi - Страница 56

The two parameters in the population of values of a quantitative variable that summarize how the variable is distributed are the parameters that measure the typical or central values in the population and the parameters that measure the spread of the values within the population. Parameters describing the central values in a population and the spread of a population are often used for summarizing the distribution of the values in a population; however, it is important to note that most populations cannot be described very well with only the parameters that measure centrality and the spread of the population.

Measures of centrality, location, or the typical value are parameters that lie in the “center” or “middle” region of a distribution. Because the center or middle of a distribution is not easily determined due to the wide range of different shapes that are possible with a distribution, there are several different parameters that can be used to describe the center of a population. The three most commonly used parameters for describing the center of a population are the mean, median, and mode. For a quantitative variable X.

 The mean of a population is the average of all of the units in the population, and will be denoted by µ. The mean of a variable X measured on a population consisting of N units is

 The median of a population is the 50th percentile of the population, and will be denoted by μ~. The median of a population is found by first listing all of the values of the variable X, including repeated X values, in ascending order. When the number of units in the population (i.e., N) is an odd number, the median is the middle observation in the list of ordered values of X; when N is an even number, the median will be the average of the two observations in the middle of the ordered list of X values.

 The mode of a population is the most frequent value in the population, and will be denoted by M. In a graph of the probability density function, the mode is the value of X under the peak of the graph, and a population can have more than one mode as shown in Figure 2.8.

The mean, median, and mode are three different parameters that can be used to measure the center of a population or to describe the typical values in a population. These three parameters will have nearly the same value when the distribution is symmetric or mound shaped. For long-tailed distributions, the mean, median, and mode will be different, and the difference in their values will depend on the length of the distribution’s longer tail. Figures 2.12 and 2.13 illustrate the relationships between the values of the mean, median, and mode for long-tail right and long-tail left distributions.

Figure 2.12 The relationships between μ,μ~, and M for a long-tail right distribution.

Figure 2.13 The relationships between μ,μ~, and M for a long-tail left distribution.

In general, the mean is the most commonly used measure of centrality and is a good measure of the typical value in the population as long as the distribution does not have an extremely long tail or multiple modes. For long-tailed distributions, the mean is pulled out toward the extreme tail, and in this case it is better to use the median or mode to describe the typical value in the population. Furthermore, since the median of a population is based on the middle values in the population, it is not influenced at all by the length of the tails.

Example 2.14

Consider the two populations that are listed below.

These two populations are identical except for their largest values, 67 and 670. For population 1, the mean is

Now, because there are 11 units in population 1, the median is the sixth observation in the ordered list of population values. Thus, the median is 28. For population 2, the mean is

Since there are also 11 units in population 2, the median is also the sixth observation in the ordered list of population values. Thus, the median of population 2 is also 28.

Note that the mean of population 2 is more than twice the mean of population 1 even though the populations are identical except for their single largest values. The medians of these two populations are identical because the median is not influenced by extreme values in a population. In population 1, both the mean and median are representative of the central values of the population. In population 2, none of the population units is near the mean value, which is 79.63. Thus, the mean does not represent the value of a typical unit in population 2. The median does represent a fairly typical value in population 2 since all but one of the values in population 2 are relatively close to the value of the median.

The previous example illustrates the sensitivity of the mean to the extremes in a long-tailed distribution. Thus, in a distribution with an extremely long tail to the right or left, the mean will often be less representative than the median for describing the typical values in the population.

Recall that a multi-modal distribution generally indicates there are distinct subpopulations and the subpopulations should be described separately. When the population consists of well-defined subpopulations, the mean, median, and mode of each subpopulation should be the parameters of interest rather than the mean, median, and mode of the overall population. Furthermore, in a study with a response variable and an explanatory variable, the mean of the subpopulation of response values for a particular value of the explanatory variable is called a conditional mean and will be denoted by μY|X. Conditional means are often modeled as functions of an explanatory variable. For example, if the response variable is the weight of a 10-year-old child and the explanatory variable is the height of the child, then the distribution of weights conditioned on height is a conditional distribution. In this case, the mean weight of the conditional distributions would be a conditional mean and could be modeled as a function of the explanatory variable height.

Example 2.15

The distribution of the age of onset of obsessive compulsive disorder given in Figure 2.14 is bimodal indicating there are possibly two subpopulations.

Figure 2.14 The distribution of the age of onset of obsessive compulsive disorder.

The mean value of the age of onset for OCD is µ = 16.2, which is not representative of a typical value in either subpopulation. OCD has two clinical diagnoses, Child Onset OCD and Adult Onset OCD, and the mean values of these subpopulations are μC=10.3 and μA=25, respectively, and clearly, the subpopulation mean values provide more information about their typical values than does the overall mean. The distributions of the Child Onset OCD and Adult Onset OCD are given in Figure 2.15.

Figure 2.15 The distributions of the age of onset of Child Onset and Adult Onset OCD.

Finally, for distributions with extremely long tails, another parameter that can be used to measure the center of the population is the geometric mean. The geometric mean is always less than or equal to the mean (µ) and is not as sensitive to the extreme values in the population as the mean is. The geometric mean will be denoted by GM and is defined as

where N is the number of units in the population. That is,

where the X values for the N units in the population are X1,X2,X3,…,XN.

Example 2.16

The distribution given below has a long tail to the right.

In a previous example, µ was computed to be 79.63. The geometric mean for this population is

Thus, even though there is an extremely large and atypical value in this population, the geometric mean is not sensitive to this value and is a more reasonable parameter for representing the typical value in this population. In fact, the geometric mean and median are very close for this population with GM = 29.4 and μ~=28.