SMHS UbiquitousVariation
Contents
Scientific Methods for Health Sciences - Ubiquitous Nature of Process Variability
Overview
In the real world, variation - which is the extent to which data from the same study may differentiate from each other - exists in almost all data sets. The truth is that no matter how controlled an environment might be in terms of its protocol or design, virtually any repeated measurement, observation, experiment, trial or study is bound to generate data that varies due to intrinsic effects (which are internal to the system being studied) and extrinsic effects (which originate from the system's ambient environment). To measure variability requires using measures that provide information on the variation of data and that illustrate the extent to which data are dispersed or spread out. We will introduce several basic measures of variation that are commonly used in statistics, including range, variation, standard deviation, sum of squares, Chebyshev’s theorem, and empirical rules.
Motivation
Variation is of significant importance in statistics and is ubiquitous in data. Consider the example in UCLA’s study of Alzheimer’s disease, which analyzed the data of 31 Mild Cognitive Impairment (MCI) and 34 probable Alzheimer’s disease (AD) patients. The investigators made every attempt to control as many variables as possible, but the demographic information they collected from the outcomes of their subjects were unavoidably varied. The same study found variation in the MMSE cognitive scores even in the same subject. The table below shows the demographic characteristics for the subjects and patients included in this study, where the following notation is used M (male), F (female), W (white), AA (African American), A (Asian).
Variable | Alzheimer’s disease | MCI | Test statistics | Test score | P-value |
Age (years) | 76.2 (8.3) range 52–89 | 73.7 (7.4) range 57–84 | Student’s T | $t_o = 1.284$ | p=0.21 |
Gender (M:F) | 15:19 | 15:16 | Proportion | $z_o = -0.345$ | p=0.733 |
Education (years) | 14.0 (2.1) range 12–19 | 16.23 (2.7) range 12–20 | Wilcoxon rank sum | $w_o = 773.0$ | p<0.001 |
Race (W:AA:A) | 29:1:4 | 26:2:3 | $\chi_{(df=2)}^2$ | $\chi_{(df=2)}^2=1.18$ | 0.55 |
MMSE | 20.9 (6.3) range 4–29 | 28.2 (1.6) range 23–30 | Wilcoxon rank-sum | $w_o= 977.5$ | p<0.001 |
Once we accept that all natural phenomena are inherently variant and that there are no completely deterministic processes, we need to look for measures of variation that allow us to know the extent to which data are dispersed. For example, suppose that we flip a coin 50 times and get 15 heads and 35 tails. According to probability theory, we have a 50% chance of getting heads or tails whenever we flip a coin, so one would expect us to end up with 25 heads and 25 tails. What happened here? Now, suppose there are 100 students and that each one flipped the coin 50 times; what would you imagine the results to be like?
Theory
Measures of variation:
- Range: The simplest measure of variation and the difference between the largest value and the smallest; Range = Maximum – Minimum.
- Suppose Jack's pulse rate varied from 70 to 76 while Tom's varied from 58 to 79. Jack has a range of 76 – 70 = 6 and Tom has a range of 79 – 58 = 21. Hence we conclude that Tom has a larger variation in pulse rate compared to Jack with the range measure.
- A similar measure of variation covers (more or less) the middle 50 percent. It is the interquartile range: Q_{3} - Q _{1} where Q _{1} and Q _{3} are the first and third quarters.
- Variance: Unlike range, which only involves the largest and smallest data, variance involves all the data values.
- Population variance: The second central moment, relative to the mean (first moment) $\mu =E(X)$: $ \operatorname{Var}(X) = \operatorname{E}\left[(X - \mu)^2 \right].$
- Unbiased estimate of the population variance (sample-variance): $s^2=\frac{\sum {(x_i-\bar{x})^2}}{n-1}$, where $\bar{x}=\frac{1}{n}\sum_{i=1}^{n}{x_i}$ is the sample mean and $n$ is the sample size.
- Standard deviation (SD): The square root of variance. Given that the deviations in variance were squared, meaning the units were squared, so to take the square root of the variance gets the unit back the same as the original data values.
- Population SD: $\sigma =\sqrt{E(X-\mu)^2}$, where $\mu$ is the population mean of the data.
- Unbiased estimate of the population stand deviation (sample standard deviation): $s=\sqrt{\frac{\sum {(x_i-\bar{x})^2}}{n-1}}$ where $\bar{x}=\frac{1}{n}\sum_{i=1}^{n}{x_i}$ is the sample mean and $n$ is the sample size.
- Consider an example: a biologist found 8, 11, 7, 13, 10, 11, 7 and 9 contaminated mice in 8 different groups. Calculate $s$. The sample average is: $\bar{x} =\frac{8+11+7+13+10+11+7+9}{8}=9.5.$
x | 8 | 11 | 7 | 13 | 10 | 11 | 7 | 9 | Sum |
$\sum {(x_i-\bar{x})}$ | -1.5 | 1.5 | -2.5 | 3.5 | 0.5 | 1.5 | -2.5 | -0.5 | 0 |
$\sum {(x_i-\bar{x})^2}$ | 2.25 | 2.25 | 6.25 | 12.25 | 0.25 | 2.25 | 6.25 | 0.25 | 32 |
- $s=\sqrt{\frac{\sum {(x_i-\bar{x})^2}}{n-1}} = \sqrt{\frac{32}{7}} \approx 2.14$
- Sum of squares (shortcuts)
- The sum of the squares of the deviations fro the means is given a shortcut notation and several alternative formulas.
$$SS(x)=\sum_{i} {(x_i - \bar{x})^2}$$
- A little algebraic simplification yields: $SS(x)=\sum_i {x_i^2} - \frac{(\sum_i{x_i})^2}{n}$.
Chebyshev’s Theorem
The proportion of the values that fall within $k$ standard deviations of the mean will be at least $1-1/k^2$ where $k$ is the number greater than 1. Chebyshev’s theorem is true for any sample set from any (well-defined) distribution. Formally, if X is a random variable from a distribution with finite mean ($\mu$) and finite non-zero variance ($\sigma^2$), then for any real number $k > 0$, $\Pr(|X-\mu|\geq k\sigma) \leq \frac{1}{k^2}$.
- The useful case is when $k \gt 1$ as when $k \lt 1$ the right-hand side is greater than one, and all probabilities are smaller than or equal to one. If $k=1$, the inequality degenerates into a trivial fact, saying that the probability is less than or equal to one. For example, using $k=\sqrt{2}$ yields that at least half of the values of this distribution lie in the (symmetric) interval $\mu − \sqrt{2}\sigma, \mu + \sqrt{2}\sigma$. Note that this is true for all distributions, whether they are symmetrical or not.
- Empirical Rule: This rule only works for bell-shaped (normal) distributions. With this kind of distribution, approximately 68% of the data values fall within one standard deviation of the mean; approximately 95% fall within two standard deviations of the mean; and approximately 99.7% fall within three standard deviations of the mean.
Applications
- An article titled The Population Genetics of Structural Variation discusses variation in the human genome. It summarizes recent dramatic advances and illustrates the diverse mutational origins of chromosomal rearrangements, arguing that their complexity necessitates a re-evaluation of existing population genetic methods. The article begins with an introduction on genomic variants, their biological significance and their basic characteristics, leading to the importance of study on structural variation. It then points out how knowledge of structural variation in the human genome has improved, and it concludes with two important challenges that will be faced by future studies of structural variation.
Software
Problems
- Let $X$ be a random variable with a mean of 80 and a standard deviation of 12. Find the mean and the standard deviation of the following variable: $Y=X-20$. Choose one answer.
- (a) Mean = 60, standard deviation = 144
- (b) Mean = 60, standard deviation = 12
- (c) Mean = 80, standard deviation = 12
- (d) Mean = 60, standard deviation = -8
- A physician collected data on 1,000 patients to examine their heights. A statistician hired to look at the files noticed the typical height was about 60 inches, but found that one height was 720 inches. This is clearly an outlier. The physician is out of town and can't be contacted, but the statistician would like to have some preliminary descriptions of the data to present when the doctor returns. Which of the following best describes how the statistician should handle this outlier? Choose one answer.
- (a) The statistician should publish a paper on the emergence of a new race of giants.
- (b) The statistician should keep the data point in; each point is too valuable to drop one.
- (c) The statistician should drop the observation from the analysis because this is clearly a mistake; the person would be 60 feet tall.
- (d) The statistician should analyze the data twice, once with and once without this data point, and then compare how the point affects conclusions.
- (e) The statistician should drop the observation from the dataset because we can't analyze the data with it.
- Researchers do a study on the number of cars that a person owns. They think that the distribution of their data might be normal, even though the median is much smaller than the mean. If they were to make a p-plot, what would it look like? Choose one answer.
- (a) It would not be a straight line.
- (b) It would be a bell curve.
- (c) It would be a group of points clustered around the middle of the plot.
- (d) It would be a straight line.
- Which of the following parameters is most sensitive to outliers? Choose one answer.
- (a) Standard deviation
- (b) Interquartile range
- (c) Mode
- (d) Median
- Which value given below is the best representative for the following data?
- Choose one answer.
- (a) The weighted average of the two modes or (4*5 + 9*5)/10 = 6.5
- (b) No single number could represent this data set
- (c) The average of the two modes or (4 + 9) / 2 = 6.5
- (d) The mean or (2 + 3 + 4 + … + 10 + 11)/18 = 5.9
- (e) The median or (6 + 7)/2 = 6.5
- The following data is collected from a website for 121 schools, and it includes these attributes about each institution: Name, Public or Private Institution, State , Cost of Health Insurance, Resident Tuition, Resident Fees, Resident Total Expenses, Nonresident Tuition, Nonresident Fees, and Nonresident Total Expenses in 2005. Some students were surprised that medical schools charged no tuition for residents, but others had to pay about $\$20,000$ in fees.
Min | Q1 | Median | Q3 | Max | |
Private | -$\$6,550$ | $\$30,729$ | $\$33,850$ | $\$36,685$ | $\$41,360$ |
Public | $\$0$ | $\$10,219$ | $\$16,168$ | $\$18,800$ | $\$27,886$ |
- On the same scale, use the 5-Number summary to construct two box plots for the tuition for residents at 73 public and 48 private medical colleges. Use the data and plots to determine which statement about centers is true.
- (a) For private medical schools, the mean tuition of residents is greater than the median tuition for residents.
- (b) With these data, we cannot determine the relationship between mean and median tuition for residents.
- (c) For private medical schools, the mean tuition of residents is equal to the median tuition for residents.
- (d) For private medical schools, the mean tuition of residents is less the median tuition for residents.
- Suppose that we create a new data set by doubling the highest value in a large data set of positive values. What statement is FALSE about the new data set? Choose one answer.
- (a) The mean increases.
- (b) The standard deviation increases.
- (c) The range increases.
- (d) The median and interquartile range both increase.
- Consider a large data set of positive values and multiply each value by 100. Which of the following statement is true? Choose one answer.
- (a) The mean, median, and standard deviation increase.
- (b) The mean and median increase, but the standard deviation is unchanged.
- (c) The standard deviation increases but the mean and median are unchanged.
- (d) The range and interquartile range are unchanged.
References
- 3D comparison of hippocampal atrophy in amnestic mild cognitive impairment and Alzheimer's disease.
- (SOCR EBook) Introduction to Variability
- Stats: Measures of Variation
- SOCR Home page: http://www.socr.umich.edu
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