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1

Definition of random variables

A variable whose observed values may be considered outcomes of an experiment and whose values cannot be anticipated with certainty before the experiment is conducted

2

What are the two types of random variables

1) Discrete variables (e.g. dichotomous, categorical)
2) Continuous variables

3

Discrete Variables

1) Can only take a limited number of values within a given range
2) Nominal: Classified into groups in an unordered manner and with no indication of relative severity (e.g., sex, mortality disease presence, race, marital status)
3) Ordinal: Ranked in a specific order but with no consistent level of magnitude of differences between ranks (e.g NYHA functional class
4) COMMON ERROR: measure of central tendency - In most cases, means and standard deviations should not be reported with ordinal data.

4

Continuous Variables

(Sometimes referred to as Counting Variables)
1) Continuous variables can take on any value within a given range.
2) Interval: Data are ranked in a specific order with a consistent change in magnitude between units; the zero point is arbitrary (e.g., degrees Fahrenheit)
3) Ratio: Like "interval" but with an absolute zero (e.g., degrees Kelvin, heart rate, blood pressure, time, distance

5

Descriptive Statistics

Used to summarize and describe data that are collected or generated in research studies. This is done both visually and numerically

6

Visual methods of describing data

1) Frequency distribution
2) Histogram
3) Scatterplot

7

Numerical methods of describing data: Measures of central tendency

a. Mean (i.e., average)
b. Median
c. Mode

8

Mean

i. Sum of all values divided by the total number of values
ii. Should generally be used only for continuous and normally distributed data
iii. Very sensitive to outliers and tend toward the tail, which has the outliers
iv. Most commonly used and most understood measure of central tendency
v. Geometric mean

SHOULD ONLY BE USED FOR NORMALLY DISTRIBUTED CONTINUOUS DATA

9

Median

i. Midpoint of the values when placed in order from highest to lowest. Half of the observations
are above and below.
ii. Also called 50th percentile
iii. Can be used for ordinal or continuous data (especially good for skewed populations)
iv. Insensitive to outliers

10

Mode

i. Most common value in a distribution
ii. Can be used for nominal, ordinal, or continuous data
iii. Sometimes, there may be more than one mode (e.g., bimodal, trimodal).
iv. Does not help describe meaningful distributions with a large range of values, each of which
occurs infrequently

11

Numerical methods of describing data

Measures of data spread and variability
a) Standard deviation
b) Range
c) Percentiles
d) Inferential Statistics

12

Standard Deviation

i. Measure of the variability about the mean; most common measure used to describe the spread
of data
ii. Square root of the variance (average squared difference of each observation from the mean);
returns variance back to original units (nonsquared)
iii. Appropriately applied only to continuous data that are normally or near-normally distributed
or that can be transformed to be normally distributed
iv. By the empirical rule, 68% of the sample values are found within ±1 SD, 95% are found within
±2 SD, and 99% are found within ±3 SD.
v. The coefficient of variation relates the mean and the SD (SD/mean × 100%).

13

Range

i. Difference between the smallest and largest value in a data set does not give a tremendous
amount of information by itself.
ii. Easy to compute (simple subtraction)
iii. Size of range is very sensitive to outliers.
iv. Often reported as the actual values rather than the difference between the two extreme values

14

Percentiles

i. The point (value) in a distribution in which a value is larger than some percentage of the other
values in the sample. Can be calculated by ranking all data in a data set
ii. The 75th percentile lies at a point at which 75% of the other values are smaller.
iii. Does not assume the population has a normal distribution (or any other distribution)
iv. The interquartile range (IQR) is an example of the use of percentiles to describe the middle
50% values. The IQR encompasses the 25th–75th percentile.

15

Inferential Statistics

1. Conclusions or generalizations made about a population (large group) from the study of a sample of that
population
2. Choosing and evaluating statistical methods depend, in part, on the type of data used.
3. An educated statement about an unknown population is commonly referred to in statistics as an
inference.
4. Statistical inference can be made by estimation or hypothesis testing.

16

Beta

- the probability of a Type II error
- the larger the Beta the lower the power of the study and the greater the chance of making a Type II error

17

Probability of NOT making a Type II error

Power = 1 - Beta

18

Type II error

- when the study states there is not difference between the groups but in reality there is a difference

19

Cost-minimization analysis

medications are considered to be equal, so just cost

20

Cost-effectiveness analysis

- Natural units (mm Hg blood pressure, blood glucose)
- Meds are not considered equal

21

Cost-benefits analysis

- Dollars
- Outcomes are expressed as benefit:cost or as net cost or net benefit

22

Cost-utility analysis

- quality of adjusted life-year (QALY)

23

Kaplan-Meier curve

- used to assess the "time to an event"
- many times this "event" is death or mortality, but in reality it can be any event

24

Cox Proportional Hazards Model

- allows the allows the investigators to control for confounding variables or factors that may be influencing the endpoint of thier study in addition to the intervention (or independent variable) being studied

25

corrrelation analysis

- generates a correlation coefficient (r) which can range from -1 to +1
- done to describe the "strength" of the relationship" between 2 variables
- the closer to +1 the stronger the "correlation" or "relationship" between 2 variables.
- it does NOT imply anything about causation
- the correlation (r) does not have the ability to determine which of the two variables came first in existence to influence the other

26

regressions analysis

- done to provide the "predictability" of one variable on another variable

27

Determining the appropriate statistical test for analyzing an endpoint

1) First ask "How many groups or samples" does this study have?
2) Second, "are these two groups (samples) related (i.e. same patient) or independent (i.e., not the same patients in both groups
3) Now determine the endpoint, "What is the endpoint"

28

Chi-square

- two independent samples
- nominal data

29

Fisher's exact

- two independent samples
- nominal data

30

Mann-Whitney U

- Two independent samples
- ordinal data
OR
- Two independent samples
- continuous data