Biostatistics Exam 3

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66 Terms

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Two sample design

  • 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

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Paired design

  • 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)

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Estimating mean difference

  • From sample of di

    • d = after-before

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Paired t-test

  • 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 (mud=0)

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

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Paired t-test statistic

  • 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

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Assumptions of paired t-test

  • Sampling units are randomly sampled from the population

  • Paired differences have a normal distribution in the population

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Formal tests of normality

  • H0: sample has normal distribution

  • HA: 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

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Shapiro-Wilk test

  • 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)

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Pooled sample variance (s2p)

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

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Two-sample t-test

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

  • Most commonly…

    • H0: mu1=mu2

    • HA: mu1 does not equal mu2

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Assumptions of two-sample t-test

  • 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

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Formal tests of equal variance

  • 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

    • HA: variances of the two groups are not equal

    • Can be extended to more than two groups

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What if variances in two groups are not equal?

  • 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

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Correct sampling units

  • 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

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Fallacy of indirect comparison

  • 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

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Interpreting overlap in confidence intervals

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

<ul><li><p>Papers often report means and confidence intervals for two or more groups without running a two-sample t-test</p></li></ul>
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t-distribution

  • 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

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One-sample t-test

  • 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

  • HA: the true mean does not equal mu0

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Interpreting t-statistic

  • 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

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Increasing sample size

  • 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)

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Assumptions of one-sample t-test

  • 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

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Estimating other statistics

  • 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

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Assumptions of calc confidence intervals for variance

  • Random sample from the population

  • Variable must have normal distribution

    • Formulas are not robust to departures from normality

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Normal quantile plot

  • 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

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Ignoring violations of normality

  • 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

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Ignoring violations of equal standard deviations

  • 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

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Data transformations

  • 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

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Log transformation

  • 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

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Other transformations

  • Arcsine (best use: proportions)

  • Square-root (counts)

  • Square (skewed left)

  • Antilog (skewed left)

  • Reciprocal (skewed right)

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Nonparametric alternatives

  • 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

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Sign test

  • 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

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Wilcoxian sign-ranked test

  • 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

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Mann-Whitney U-Test

  • 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

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Assumptions of nonparametric tests

  • 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

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ANOVA

  • 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…mun

  • HA: 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: MSgroups/MSerror = 1

    • False null: MSgroups/MSerror > 1

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Group mean square (MSgroups)

  • Proportional to the observed amount of variance among group sample means

    • Variation among groups

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Error mean square (MSerror)

  • Estimates the variance among subjects that belong to each group

    • Variation with groups

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ANOVA calculations

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

  • Grand mean

    • Equal to a’ constant

  • Group mean square (MSgroups) is variation among

  • Group error square (MSerror) 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

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Variation explained

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

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Assumptions with variation

  • 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

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Alternatives with variation

  • 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

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Planned comparisons

  • 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

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Unplanned comparisons

  • 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

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Tukey-Kramer method

  • 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

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Kruskal-Wallis post-hoc test

  • 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

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Dunn’s 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

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Correlation

  • When two numerical variables are associated then they are. correlated

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Correlation coefficient

  • 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

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Standard error of correlation coefficient

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

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Approx. confidence. interval of correlation coefficient

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

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Correlation assumptions

  • Random sample from the population

  • Bivariate normal distribution

    • Bell shaped in two dimensions rather than one

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Deviations from bivariate normality

  • Transform data (both variables same way)

  • Nonparametric test (Spearman’s rank correlation); skipping

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ANOVA as a linear model

  • 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

  • HA: 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

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Two-factor outcomes

knowt flashcard image
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Assumptions of two-factor ANOVA linear models

  • 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

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Correlation vs regression

  • 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

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Linear regression

  • 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

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Calculating intercept

  • 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

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Samples vs populations

  • 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

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Predicting values

  • 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”

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How well do data fit line?

  • 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 (MSresidual) quantifies the spread of the scatter

    • Residual mean square

    • Analogous to error mean square in ANOVA

  • Used to quantify the uncertainty of the slope

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Two types of predictions

  • 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

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ANOVA (F) approach

  • Recall two source of variation in ANOVA

    • Among groups (MSgroups)

    • Within groups (MSerror)

  • In regression framework:

  • Deviations between the predicted values Yi and Y

    • Analogous to MSgroups

  • Deviations between each Yi and its predictive value Yi

    • Analogous to MSerror

  • 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 = SSregression/SStotal

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Regression toward the mean

  • 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

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Assumptions of linear regression

  • 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

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Detecting issues

  • 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 (Yi - Yi) is plotted against Xi

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