Biostatistics Exam 3

Two groups

Each treatment group composed of independent, random sample units

Wild type vs. control, drug vs. placebo, where treatments are applied to separate and independent samples

Two groups

Each sampled unit receives both treatments

Both treatments applied to every sampled unit

More powerful b/c control for variation among sampling units

Not as common

Two measurements from same sampling units —> converted to single measurement by taking difference between them

Examples (patient weight before and after hospitalization, effects of sunscreen on one arm vs placebo on another arm, effects of environment on identical twins raised under different socioeconomic conditions)

From sample of d_i

• d = after-before

Used to test the null hypothesis that the mean difference of paired measurements equals a specific value

H₀: mean change in antibody production after testosterone implants was 0 (mu_d=0)

H_A: mean change in antibody production after testosterone implants was not 0

One paired samples reduced to single measurement (d)

Calculation is same as one-sample t-test

Can determine P with computer or statistical table

Fail to reject the null hypothesis that the mean change in antibody production after the testosterone implant is 0

Sampling units are randomly sampled from the population

Paired differences have a normal distribution in the population

H₀: sample has normal distribution

H_A: sample does not have normal distribution

Should be used with caution

• Small sample sizes lack power to reject a false null (Type II error)

• Large sample sizes can reject null when the departure from normality is minimal and would not affect methods that assume normality

Evaluates the goodness of fit of a normal distribution to a set of data randomly sampled from a population

Most commonly used formal test of normality

Estimates mean and standard deviation using sample data

Tests goodness-of-fit between sample data and normal distribution (with mean, sd of the sample)

The average of the variances of the samples weighted by their degrees of freedom

Simplest test to compare the means of a numerical variable two independent groups

Most commonly…

• H₀: mu₁=mu₂

• H_A: mu₁ does not equal mu₂

Each of two samples is a random sample from its population

Numerical variable is normally distributed in each population

• Robust to minor deviations from normality

• Need to run Shapiro-Wilk test on both samples

Standard deviation (and variance) of the numerical variable is the same in both populations

Robust to some deviation from this if sample sizes of two groups are approximately equal

An F-test is sometimes used, but it is highly sensitive to departures from the assumption that the measurements are normally distributed in the population

Levene’s test performs better and is recommended

• H₀: variances of the two groups are equal

• H_A: variances of the two groups are not equal

• Can be extended to more than two groups

Standard t-test works well if both sample sizes are greater than 30 and there is less than 3 fold difference in standard deviations

Welch’s t-test compares the means of two groups and can be used even when the variances of the two groups are not equal

• Slightly less power compared to standard t-test

• Should be used when the sample standard deviations are substantially different

• Formulae are different than standard two-sample t-test

When comparing the means of two groups, an assumption is that the samples being analyzed are random samples

Often, repeated measurements are taken on each sampling unit

Makes the identification of independent units more challenging

Compare each group mean to hypothesized value rather than comparing group means to each other

Comparisons between two groups should always be made directly, not indirectly by comparing both to the same hypothesized value

Ex: since group 1 is significantly difference than zero, but group 2 is not, then groups 1 and 2 are significantly different from each other

Papers often report means and confidence intervals for two or more groups without running a two-sample t-test

Similar to standard normal distribution (Z), but with fatter tails

As the sample size increases the t distribution becomes more like the standard normal distribution

Has critical values like Z

Get values with computer or table

Compares the mean of a random sample from a normal population with the population mean proposed in a null hypothesis

H₀: the true mean equals mu₀

H_A: the true mean does not equal mu₀

Compute P-value: probability of this t-statistic (or more extreme) given the null hypothesis is true

Using a stats table

• Look up critical t-value

• Observed value is within range of -/+ critical value

• Data consistent with true null

Increasing sample size reduces standard error of mean

• Uncertainty of estimate of mean

Larger sample sizes increase probability of rejecting a false null hypothesis (power)

If this null is really false, then the sample of 25 failed to detect a false null (Type II error)

Data are a random sample from the population

Variable is normally distributed in the population

• Few variables in biology are exact match to normality

• But in many cases the test is robust to departures from normality

Emphasis on estimating the mean of a normal population

Spread of the sample distribution (standard deviation or variance)

Confidence limits for variance is based on the X² distribution

Random sample from the population

Variable must have normal distribution

• Formulas are not robust to departures from normality

Compares each observation in the sample with its quantile expected from the standard normal distribution. Points fall roughly along a straight line if the data come from a normal distribution

t-tests assume data are drawn from a population have a normal distribution

But can sometimes be used when data are not normal

• Central limit theorem

When sample sizes are large the sampling distribution of means behaves roughly as assumed by t-distribution

Large sample size depends on shape of the distribution

• If distributions of two groups being compared are skewed in different directions, then avoid t-tests even for large samples

• If distributions are similarly skewed then there is more leeway

Two sample t-tests assume standard deviations in the two populations

If sample sizes are >30 in each group AND sample sizes in two groups are even, then even up to a 3x difference can be ok

Otherwise, use Welch’s t-test

A data transformation changes each measurement by the same mathematical formula

Can make standard deviations more similar and improve fit of the normal distribution to the data

All observations must be transformed

If two samples then they both must be transformed in same way

If used then usually best to back transform confidence intervals

Data are transformed by taking the natural log (ln) or sometimes log base-10 of each measurement

Common uses:

• Measurements are ratios or products

• Frequency distribution skewed to the right

• Group having larger mean also has larger standard deviation

• Data span several orders of magnitude

Arcsine (best use: proportions)

Square-root (counts)

Square (skewed left)

Antilog (skewed left)

Reciprocal (skewed right)

A nonparametric method makes fewer assumptions than standard parametric methods do about the distributions of the variables

• Can be used when deviations from normality should not be ignored, and sample remains non-normal even after transformation

Do not rely on parametric statistics like mean, standard deviation, variance

Usually based on ranks of the data points rather than the actual values

Compares the median of a sample to a constant specified in the null hypothesis. It makes no assumptions about the distribution of the measurement of the population

Each measurement is characterized as above (+) or below (-) the null hypothesis

If the null is true, then you expect the half the measurements to be + and half to be -

Uses binomial distribution to test if the proportion of measurements above the null hypothesis is p = 0.5

More power than standard sign test because information about the magnitude away from the null for each data point

But test assumes that population is symmetric around the median (i.e., no skew)

Nearly as restrictive as normality assumption, thus not recommended

Nonparametric test for two samples

Compares the distributions of two groups. It does not require as many assumptions as the two-sample t-test

What if you have tied ranks?

• Assign all instances of the same measurement the average of ranks that the tied points would have received

Mann-Whitney U-test further explained from office hours:

inability to do two sample t-test due to lack of normal distribution

e.g. right skew failed shapiro test for group 1 and group 2, but group 2’s distribution looks a bit different

takes all data points and ranks them from low to high

• null is that distribution of ranks is equal in group 1 and group 2

  • sprinkling of black and green along the line would be equal visually

e.g. now imagine that right skew for group 1 and left skew for group 2

• lowest is only black then some green, then you get to the highest and its mostly/all green with little black

• this would show the alternative hypothesis that distribution of ranks is not equal

  • p<0.05

Still assume that both samples are random samples from their populations

Wilcoxian signed-rank test assumes distributions are symmetrical (big limitation-not recommended)

Rejecting null hypothesis of Mann-Whitney U-test means two groups have different distributions of ranks, but does not necessarily imply that means of medians of groups differ

To make this inference there is an assumption that the shapes of the distributions are similar

Analysis of variance (ANOVA) compares the means of multiple groups simultaneously in a single analysis

• Tests for variation of means among groups

H₀: mu₁ = mu₂ = mu₃…mu_n

H_A: mean of at least one group is different from at least one other group

Null assumption that all groups have the same true mean is equivalent to saying that each group sample is drawn from the same population 

But each group sample is bound to have a different mean due to sampling error

ANOVA determines if there is more variance among sample means than we would expect by sampling error alone

• Two measures of variation

Test statistic is a ratio:

• True null: MS_groups/MS_error = 1

• False null: MS_groups/MS_error > 1

Proportional to the observed amount of variance among group sample means

• Variation among groups

Estimates the variance among subjects that belong to each group

• Variation with groups

Sums of squares (SS) calculates two sources of variation (among and within groups)

Grand mean

• Equal to a’ constant

Group mean square (MS_groups) is variation among

Group error square (MS_error) is variation among individuals in same group

F-ratio test statistic

• Has pair of degrees of freedom

  • Numerator and denominator

• Use to calculate P-value with stats table or computer

R² measures the fraction of variation in Y that is explained by group differences

Measurements in every group represent a random sample from the corresponding population

Variable is normally distributed in each of the k populations

• Robust to deviations, particularly when sample size is large

Variance is the same in all k populations

• Robust to departures if sample sizes are large and balanced, and no more than 10x differences among groups

Test normality with Shapiro-Wilk and test equal variances with Levene’s test

Data transformations can make data more normal and variances more equal

Nonparametric alternative: Kruskal-Wallis test

• Similar principle as Mann-Whitney U-test

A planned comparison is a comparison between means planned during the design of the study, identified before the data are examined

In circadian clock follow-up study, the planned (a priori) comparison was difference in means between knee and control group

Comparisons are unplanned if you test for differences among all means

Problem of multiple tests (increasing probability of Type I error) should be accounted for

With the Tukey-Kramer method the probability of making at least one Type I error throughout the course of testing all pairs of means is no greater than the significance level

Works like a series of two-sample t-tests, but with a higher critical value to limit the Type I error rate

• Because multiple tests are done, the adjustment makes it harder to reject the null

Suppose that your data…

• Fail normality even after transformation

• Generate a significant Kruskal-Wallis result

So the interpretation is that the distribution of ranks differs for at least one group. But which one?

Should not use Tukey-Kramer, which is a parametric test

The appropriate analysis for a post-hoc analysis of groups following a significant Kruskal-Wallis result

• Will compare all possible pairs of groups while controlling for multiple tests

When two numerical variables are associated then they are. correlated

The correlation coefficient measures the strength and direction of the association between two numerical variables

• AKA linear correlation coefficient or Pearson’s correlation coefficient

Correlation coefficient (statistic), r

Population correlation coefficient (parameter), p

Ranges from -1 to 1

Possible that two variables can be strongly associated but have no correlation (r=0)

• Non-linear association

Sampling distribution of r is not normally distributed, so SE_r isn’t used in calculating the 95% CI

Involves conversion of r that includes natural log, and then back conversion

Random sample from the population

Bivariate normal distribution

• Bell shaped in two dimensions rather than one

Transform data (both variables same way)

Nonparametric test (Spearman’s rank correlation); skipping

Linear model Y=+A

• Y: response

• : grand mean (constant)

• A: treatment

Circadian clock study: SHIFT = CONSTANT + TREATMENT

H₀: treatment means are all the same

• SHIFT = CONSTANT

H_A: treatment means are not all the same

• SHIFT = CONSTANT + TREATMENT

Even if the null hypothesis is true, however, the treatment means will be different due to sampling error

Thus, the full model (with TREATMENT) will be a better “fit” to the data

The F-ratio is used to test whether including the treatment variable in the model results in a significant improvement in the fit of the model to the data

• Compared with the fit of the null model lacking the treatment variable

ANOVA linear model: RESPONSE = CONSTANT + EXPLANATORY

Extending for multiple explanatory variables

• RESPONSE = CONSTANT + EXP1 +EXP2 + EXP1 * EXP2

• Design is called a two-way ANOVA or two-factor ANOVA

The measurements at every combination of values for the explanatory variables are a random sample from the population of possible measurements

The measurements for every combination of values for the explanatory variables have a normal distribution in the corresponding population

The variance of the response variable is the same for all combinations of the explanatory variables

Correlation measures the aspects of the linear relationship between two numerical variables

Regression is a method that predicts values of one numerical variable from values of another numerical variable

Fits a line through the data

• Used for prediction

• Measures how steeply one variable changes with the other

The most common type of regression

• Although there are non-linear models (e.g., quadratic, logistic)

Draws a straight line through the data to predict the response variable (Y, vertical axis) from the explanatory variable (X, horizontal axis)

Fitting the “best” line

• You want a line that gives the most accurate predictions of Y from X

• Least-squares regression: line for which the sum of all the squared deviations in Y is the smallest

Formula for the line

• Y = a + bX

  • a is the Y-intercept; b is the slope

  • The slope of a linear regression is the rate of change in Y per unit X

  • Also measures direction of prediction

    • Positive: as X increases Y increases

    • Negative: as X increases Y decreases

Once slope is calculated, getting intercept is straightforward because the least-squares regression always goes through point (X, Y)

• Plug mean values into line formula: Y = a + bX

• Rearrange to solve for intercept: a = Y - bX

The slope (b) and intercept (a) are estimated from a sample of measurements, hence these are estimates/statistics

The true population slope () and intercept () are parameters

Regression assumes that there is a population for every value of X, and the mean Y for each of these populations lies on the regression line

Now that you have the regression line you can predict values of Y for any specified value of X

Predictions are mean Y for all individuals with value X

Designated Y, or “Y-hat”

The residual of a point is the difference between its measured Y value and the value of Y predicted by the regression line

Residuals measure the scatter of points above and below the least-squares regression line

Can be positive or negative

Variance in residuals (MS_residual) quantifies the spread of the scatter

• Residual mean square

• Analogous to error mean square in ANOVA

Used to quantify the uncertainty of the slope

Predict mean Y for a given X

• E.g., what is the mean age of all male lions whose noses are 60% black?

Predict single Y for a given X

• E.g., how old is that lion over there with a 60% black nose?

Both predictions give the same value of Y, but they differ in precision

• Can predict mean with more certainty than a single value

Confidence bands measure the precision of the predicted mean Y for each value of X.

Prediction intervals measure the precision of the predicted single Y-values for each X

Recall two source of variation in ANOVA

• Among groups (MS_groups)

• Within groups (MS_error)

In regression framework:

Deviations between the predicted values Y_i and Y

• Analogous to MS_groups

Deviations between each Y_i and its predictive value Y_i

• Analogous to MS_error

Using ANOVA approach will generate the same P-value as the t-test approach

Can be used to measure R^2: the fraction of the variation in Y that is “explained” by X

R^2 = SS_regression/SS_total

Regression toward the mean result when two variables measured on a sample of individuals have a correlation less than one. Individuals that are far from the mean for one of the measurements will, on average, lie closer to the mean for the other measurement

Cholesterol measurements before and after drug

Solid line: linear regression

Dashed line: one-to-one line with slope of 1

At each value of X:

• There is a population of Y-values whose mean lies on the regression line

• The distribution of possible Y-values is normal (with same variance)

• The variance of Y-values is the same at all values of X

• The Y-measurements represent a random sample from the possible Y-values

Outliers

If only one (or a low number) then it may be reasonable to report regression with and without outlier

Nonlinearity can be detected by inspecting graphs

Non-normality and unequal variances can be inspected with a residual plot

Residual plot: residual of every data point (Y_i - Y_i) is plotted against X_i

If assumptions of normality and equal variances are met then there should be a roughly symmetric cloud above/below horizontal line at 0