Biostats Final Exam

Biostatistics (def)

Analysis of data derived from biological sciences and medicine

Statistical tests to analyze biological data

Describing (large/small/variable a trait is)

Testing (whether two things are related/different)

Predicting (whether an intervention is effective)

Categorical data (def, types)

Non-ordered (nominal): yes/no

Ordered (ordinal): ranked, agree/disagree/neutral

Nominal data (def, ex)

Categorical

Non-ordered, non-numerical, can’t be ranked

Yes or no

Eye color

Ordinal data (def, ex)

Categorical

Natural rank or order, but distances between are not known or equal

Poor, average, good

Education levels

Military ranks

Numeric data (def, types)

Scale

Discrete: number of siblings

Continuous/interval: height, BMI

Discrete data (def, ex)

Numeric

Countable, distinct values that can’t be broken down into smaller parts

Number of students

Clicks on a website

Continuous/interval data

Numeric

Values that can be infinitely divided, measured

Height

Weight

Temperature

Measured, not counted

Rank (def)

Relative position of a set of measurements

Rate (def, ex, notes)

Ratio of two quantities

Percentages, proportions, ratios

Zero as lowest meaningful point

Descriptive statistics (use, ex)

To present and describe data in a useful way

Show patterns and associations

Summary statistics, graphs

Inferential statistics (use, ex)

To draw conclusions about a population from a sample; taking action based on the data

Estimation

Hypothesis tests, statistical modelling

Box and whisker plot (type, notes)

Median, range, IQR, outliers

Numeric data

25% in each whisker

50% in the box

Presenting categorical data

Table

Bar chart

Pie chart (<6 slices)

Presenting numeric data

Histograms

Box and whisker plots

Mode (def, features)

Value that occurs most frequently

Useful for discrete or categorical data

Median (def, features)

Middle value

Useful for asymmetric or skewed data or outlying values

Mean (def, feat)

Average of data set (add up and divide by number of points)

Affected by extreme values

For symmetrical distributions

Fictitious

Only for interval data

Skew (left vs right)

Left peak: POSITIVE skew

Right peak: NEGATIVE skew

Lefty win :)

Range (def, feat)

Minimum and maximum values

Sensitive to extreme values

Interquartile range (def, feat)

Difference between 25th and 75th centiles

Range of middle 50% of data

Not influenced by extreme values; ok for skewed

Variance (def, feat)

Mean of the squared distances to the mean

Are data close to the mean or spread out?

Standard deviation (def, feat)

Square root of the variance

Symmetric data

Coefficient of variation (use, calc)

Compare different scales or measurements

Standard deviation divided by the mean; usually multiplied by 100 to get a percentage

P (def)

Central or expected value for a given outcome, proportion or probability

N (def)

Sample size

Variance and standard error (calc)

p(1-p)/n

Standard error is same but square rooted

Probability (def, calc)

Frequency of event in many many trials

Categorical variables

Number of events/total number of trials

Probability of two events (calcs)

Mutually exclusive: Add probabilities

Not sequential: Multiply probabilities

Binomial probabilities (def, notes)

Chance of getting a specific number of “successes” in a fixed number of independent trials

Only two outcomes (success/failure)

Constant success probability

Conditions for binomial probabilities

BINS

Binary outcomes

Independent trials

Number of trials (N)

Same probability of success

Normal distribution

Probability of a range of variables is represented by area under the curve, = 1

Mean/median/mode equal

Symmetrical around the mean

Standard deviation Normal distribution (percents)

68% +- 1 SD of mean

95% within 2

99.7% within 3

Normal distribution mean and SD

Mean (μ) = 0

SD (σ) = 1

Z value (def, equ)

Any Normal variable can be standardized to get a z-value

(value-mean value)/SD

Kurtosis (def, names)

How tailed/peaked the distribution is

High kurtosis = leptokurtic = fat tails and sharper peaks, more outliers

Low kurtosis = platykurtic = thinner tails, flatter peak, fewer outliers

Mesokurtic similar to Normal

Population and Sample (defs, issue)

Population: group of interest

Sample: group chosen to represent population

Issue: sample may not reflect population; repeated samples from same population may give different results

Population parameters (def)

What is of interest to determine

Sample estimates = sample statistics, point estimates of population parameters

Conventional notation

Population:

N = population size

μ = mean

σ = SD, with 2 = population variance

Sample:

n = sample size

x̄ = mean

sd = sample SD

sd2 = sample variance

Sampling distribution of the mean (def)

Mean and standard deviation of the sample means

Standard error of the mean (def, equ)

Standard deviation of the sampling distribution of the means

Used to comment on the population mean, not to describe the dispersion of values around a mean

SEM = sd/square root of n

Confidence intervals (def, equ)

Range of values within which the the statistic would fall 95% of the time

Indication of how good the sample mean is as an estimate of the population mean

Mean +- 1.96*SEM (for 95% CI)

Z score requisites

Random samples

Quantitative data

Variable Normally distributed

Sample size > 30

Z score interpretation

Positive Z = observation is >mean

Negative z = observation is <mean

Sampling distribution of a proportion (def, notes)

the distribution of all possible values of the proportion that could be obtained in repeated samples of the same size from the population of interest

has a shape, mean and SD

mean = π

Standard error of proportion

Standard deviation of the sampling distribution of the proportion

Gets smaller as sample size increases

Central limit theorem

Large enough sample size 30+ → distribution of sample means will approximate a normal distribution

Finite population correction factor

1-f

sampled most or all of the population

Poisson distribution (use, ex)

Model rare events that occur across time, i.e. 100 year floods

Less than 20 events

Rare health events like infant mortality, cancer

Confidence intervals for an age-adjusted rate

SE = R/sqaure root of N

R = age-adjusted rate

N = number of events (deaths)

CI: R+-1.96xSE

Pnorm

Converts a Z score into a probability

Find probability that a randomly selected value from a Normal distribution would be less than or equal to a specified value

Output = cumulative density function, area under the curve to the left of a Z score

Qnorm

Converts probability into a Z score

Find the specific value in a Normal distribution at which a given proportion of the distribution falls before

Output = quantile or value below which the given percentage of the distribution lies

Hypothesis (def)

A test of belief or set of rules from coming to a yes/no conclusion

Statistical hypothesis = statement of belief regarding the value of one or more population characteristics

Hypothesis (process)

State null hypothesis (Ho) → there is NO EFFECT

Apply a statistical test

Decide to accept or reject null hypothesis (p-value)

Interpret the test

T test (use, assumptions)

To test if two means are the same

One continuous, one categorical

Assumes:

• distributions roughly symmetrical

• Observations independent

• Variances are similar

T statistic/value (def, equ)

How far the data are from the null hypothesis

A standardized difference in means

(mean 1 - mean 2)/SE of mean diff

T score (meaning, equ)

How many SEMs are we from the zero?

A standardized score for comparison

P value (def, threshold)

Probability of the observed test statistic if null is true

P>0.05, accept the null

P<0.05, reject the null

Cohen’s d (def, interp)

How many SDs separate the means of two groups

Standardized measure of effect size

Small = .2

Medium = .5

Large = .8

Effect size for t tests

Type I error

False positive

Rejecting the null when it’s true

Controlled by significance level α, usually .05 or 5%

Type II error

False negative

Failing to reject null when it’s false

Controlled through large enough sample size

β

Power of a test (def, equ)

Probability of correctly rejecting a null

1-β

Usually .2 or power 80%

Paired t test (use)

Need for a paired sample i.e. a pre- and post-treatment measurement on the same participants

Calculates the difference within pairs (change score) first (subtract pre from post), and take the mean

ANOVA

Analysis of variance'/how different the means are

T test for 2+ means

Alt: at least one mean is different

Effect size: Eta squared or partial eta squared

Between-group differences (ANOVA)

Using group means and grand mean

Sums of squared deviations, then mean squared deviations (between)

Within-group differences

Using raw values and their group mean

Sum of squared deviations, then mean squared deviations (within)

Mean squares (ANOVA, calc)

= Sums of squares/degrees of freedom

Degrees of freedom (ANOVA)

Between groups (df1) = number of groups - 1

Within groups (df2) = N total - number of groups

Total df = N total - 1

ANOVA interpretation

If significant (p>0.05), justification to go look at differences between groups to find weirdo using Tukey’s HSD post-hoc

ANOVA post-hoc tests (why, ex)

To control for type I error at 5% level

Usually Tukey’s HSD

Partial eta squared (what, which test, interp)

ANOVA

Proportion of the variance in outcome variable explained by the grouping factor

Small effect: 0.01

Medium: 0.06

Large: 0.15

ANOVA assumptions

Groups normally distributed

Groups have same variance (up to 2x ok)

Observations and groups independent

Ok with moderate violations of assumptions (robust) but worse if groups very different sizes/small sample sizes

Correlation (def)

Measures association between two CONTINUOUS numeric variables

Association not causation

Assumes a straight line is the true relationship

Pearson’s r (def, interp, assum)

Correlation coefficient

Ranges from -1 to 1

<0.5 = weak

.5-.7 = moderate

>.7 = strong

Assumes Normally distributed variables, straight line relationship

Scatterplot (set up)

Dependent variable on the Y

Predictor on the X

Spearman’s ρ and Kendall’s τ

Alternate correlation coefficients for non-Normal or ordinal variables

Non-parametrics

R2 (R squared) (def, use)

Square of Pearson’s r

Effect size for correlation; amount of overlap between the variables

If r2 = 0.25, 25% of the variance is shared between variables

Uses of correlation (3)

Association between two variables in observational studies

Validating a new test against a gold standard

Reliability of a test

Null hypothesis for correlation

There is no correlation, r = 0

Linear regression (def, equ)

Gives equation of the line of the association

y = mx+b or y=b0+b1x

m or b1 is slope

b or b0 is intercept

Interpreting the slope

Slope quantifies how different y is for a +1 unit difference in x

Least squares (def, equ)

Regression fits the “best line” where the distance squared from each data point to the line is kept as small as possible

slope = SSyx/SSx

Error/residuals in regression (def)

The distances from each point to the line (vertically)

Represents unexplained variability in Y

Software output of note for regression (4)

R2 value

Standard error of the estimate

Coefficients: Intercept and Slope, including P-values

Assumptions of linear regression (3)

Linear relationship (check using scatterplots)

Constant variance of the residuals (no wedge shape)

Residuals have Normal distribution

Stability of a regression line (depends on)

Sample size

<5 observations per predictor is unstable

10-15 at least, >20 ideally

100 plus the number of predictor variables

Chi-square test (use, output, null)

Compare proportions between two or more groups

Categorical and categorical

Give x2 (chi-squared) test statistic

Null = proportions in groups are the same; being active or not is independent of gender

Expected table (chi) (def, equ, assum)

What you would expect in the cells of a table if the null were true

= row total x column total/grand total

Should have N>=20 and no cell <=5 unless N>=40; if no, FIsher’s

Chi-squared degrees of freedom (how, equ)

Depends on size of table

df = (#rows-1)x(#columns-1)

Assumptions of chi-square test (3, additional/alternate tests)

Sample size large enough (Fisher’s exact test)

Independent data points, not pre-post (McNemar’s tests)

Distribution curve is continuous, while cell counts are discrete (Yate’s continuity correction?)

Complete follow-up (def, notes)

All participants followed to death

Length of survival known for everyone

Rare and difficult; data is often “censored” or lost to follow-up

Right-censored data (def)

Blind to after

We don’t know what happened after a particular time or when a future event happens

Left-censored data (def)

Blind to the past

We don’t know what happened before a particular time or when a past event happened

Variable time follow-up (def)

Participants aren’t followed to death

Analyzing variable-time studies (best to worst, 4)

Life table approach (optimum)

Using person-years as a unit of observation (acceptable but unrealistic)

Comparing N-year risk (biased if any incomplete FUP)

Comparing mean survivals (don’t work)

Clinical/actuarial life table (def, equ)

Gives survival probabilities over time

Probability of surviving from time 0 to time b = (prob of surviving 0 to a)*(prob of surviving a to b)

Kaplan Meier approach (def)

Shows survival probabilities over time (steps)

Splits periods at events (outcome aka death, withdrawal, etc)

Log-rank test (use)

Compare two+ groups for survival time

Median survival time (def, note)

Time point where 50% survival is reached

Only estimable when curves drop below 50% survival

Exploratory study objectives (3)

State of low knowledge

Baselines, natural history

Discovery of patterns/potential assocations

Confirmatory study objectives (3)

Find effect of a given magnitude

Control of type I and II errors

Possible population state/hypothesis test