1. discrete (dichotomous, categorical)
2. continuous

1. Nominal
2. Ordinal

can only take a limited number of values within a given range

Discrete variable

Classified into groups in an unordered manner &  with no indication of relative severity (sex, mortality, disease state)

discrete variable

• ranked in specific order
• no consistent level of magnitude of difference between ranks (NYHA functional class describes the functional status of pts w/heart failure & subjects classified in increasing order of disability I, II, III, IV)
• means & SD should not be used with ordinal data in most cases (measure of central tendency)

Ranked in a specific order with a consistent change in magnitude between units

Zero point is arbitrary (e.g. Fahrenheit)

Continuous variable

• Interval
• Ratio

• Can take on any value within a given range
• Sometimes referred to as Counting Variable

Data ranked in a specific order with a consistnet change in magnitude between units

Has an absolute zero (e.g. degrees Kelvin, pulse, BP, time, distance)

Continuous variable

1. Mean
2. Median
3. Mode

Presenting dat using only measures of central tendency can be misleading without some idea of data spread

• Average
• Measure of central tendency
• Sum of all values divided by the total number of values
• Generally used only for continuous & normally distributed data
• Very sensitive to outliers  & tend toward the tail which has the outliers 

Geometric mean

• Measure of central tendency
• Midpoint of the values when placed in order from highest to lowest
• half of the observations are above & half below
• Also called the 50th percentile
• Can be used for ordinal or continuous data
• especially good for skewed populations
• Insensitive to outliers

Mode

• Most common value in a distribution
• Measure of central tendency
• can be used for nominal, ordinal or continuous data
• Can be more than 1 mode: bimodal & trimodal
• Does not help describe meaningful distributions with a large range of calues, each of which occur infrequently

1. Standard deviation
2. Range
3. Percentiles

• Meaure or the variability about the mean (most common measure used to describe this)
• Square root of the variance (average squared difference of each observation from the mean) returns variance back into original units (non-squared)
• apply only to continuous data normally or near-normally distributed
• 68% within 1 SD
• 95% within 2 SD
• 99% within 3 SD

relates the mean & SD

SD/mean x 100%

• Difference between the smallest and largest values
• does not give a tremendous amount of info by itself
• Easy to compute (simpel subtraction)
• Size of range is very sensitive to outliers
• Often reported as the actual values rather than the difference between the 2 extreme values

• The point (value) in a distribution in which a value is larger than some % of the other values in the sample
• 75^th percentile (75% of the values are smaller)
• Does not assume the population has a normal distribution
• IQR: middle 50% - encompasses the 25^th to the 75^th percentile

Discrete

• Binomial
• Poisson

Normal (Gaussian)

•  Most common model for population distribution
• Symetric or "bell-shaped" frequency distribution
• median and mean will be approx. equal
• Frequency ditribution & histograms (visual check look symmetrical & bell-shaped)
• formal test: Kolmogorov-Smirnov

Landmarks for continuous, normally distributed data

1. m: population mean is equal to zero
2. s: population SD is equal to 1
3. x and s represent the smple mean and SD

• quantifies uncertainty in the estimate of the mean NOT variability in the sample
• estimates the SD of the means

• 95% CI most common
• gives and idea of the magnitude of the difference between groups as well as statistical significance
• if CI contains 0 = no difference between 2 variables (interpreted as not statistically significant: a p-value _^>/= 0.05)
• no need to show both 95% CI & p-value

odds ratio & relative risk:

value of 1 indicates no difference in risk, no statisical difference

H_o (Null Hypothesis)

(rejected or not rejected) 

no difference between groups being compared

(treatment A = treatment B)

•  if rejected: statistical significance between groups (unlikely attributable to chance)
• if accepted (not rejected): no statistically significant difference between groups (any apparent differences may be attributable to chance) not concluding that they are equal

• opposite of null hypothesis
• States that there is a difference between groups

Parametric Tests

Assumptions

Examples (3)

• Assume: data normal or near normal distribution - evaluate whether this is true by comparing mean to median
• data is continuous
• measured on an interval or ratio scale

Examples:

student t-test:::

one-sample test: compares mean of the study with population mean

two-sample test: compares means of 2 independent samples

(may not use multiple t-tests with more than2 groups)

Paired test:::

compares the mean difference of paired or matched samples

Parametric tests

analysis of variance

1. One-way ANOVA: compares means of 3 or more groups
2. Two-way ANOVA: additional factor (e.g. age) added
3. Repeated-Meaures ANOVA: a related samples test

Nonparametric Tests

• may also be used for ordinal & continuous data that do not  meet the assumptions of the t-tests or ANOVA

Tests:

1. Wilcoson rank sum & Mann-Whitney U-test, compares 2 independent samples (related to a t-test)
2. Kruskal-Wallis one-way ANOVA by ranks, compares 3 or more independent groups (related to a one-way ANOVA) - - post hoc testing

Nonparametric Tests

(tests for related or paired samples)

1. Sign test & Wilcoson signed rank test: compares 2 matched or paired samples
2. Firedman ANOVA by ranks: compares 3 or more matched/paired groups

Chi-square (x²) test: compares expected & observed proportions between 2 or more groups:   

-   test of independence

-   test of goodness of fit

Fisher exact test: specialized version of chi-square test for small groups (cells) containing < 5 observations

McNemar: paired samples                           

Mantel-Haenszel: controls for the influence of confounders

H_o is rejected, when it should not have been

- conclude that there is a statistically significant difference when actually one does not exist

termed a - convention is to set it at 0.05 (5%)

H_o is not rejected, but should have been

- concluding that no differnce exists when one truly does

termed b  - conventionally set 0.20 to 0.10

1-b

• the probability of making a corret decision when H_o is false
• the ability to detect differences between groups if 1 actually exists

Dependent on:

-predetermined a: the risk of error you will tolerate when rejecting H_o

- sample size

- the size of the difference between the outcomes wanting to detect

- variability of the outcomes being measured