NOTES NOTES PARAMETRIC TESTS PARAMETRIC TESTS ▫ Independent observations ▫ Population standard deviations (SDs) are same ▫ Data distributed normally/approximately normally ▪ ANOVA, t-tests ▪ Use for following data ▫ Randomly selected samples ANOVA osms.it/one-way_ANOVA osms.it/two-way_ANOVA osms.it/repeated-measures_ANOVA ▪ AKA analysis of variance ▪ Determines differences between > two samples ▫ Measures differences among means ▪ F-ratio (F statistic) ▫ F = (variance between groups) (variance within each group) ▪ Computer program calculates p-value from F; use F to accept/reject null hypothesis ▫ F approx. = 1; p large; accept null hypothesis ▫ F large → p small (alpha set at 0.05 signiﬁcant → reject null hypothesis) ▪ Assumptions ▫ Samples drawn randomly; sample groups have homogeneity of variance (i.e. from same population; interval, ratio data) 1-way ANOVA ▪ Between groups design ▪ One independent variable ▫ May have multiple levels (e.g. drug A effect vs. drug B vs. placebo on speciﬁed outcome) Factorial ANOVAs ▪ Factorial designs ▪ Two-way, three-way, four-way ANOVA, more (two, three, four, etc. independent variables) Single-factor repeated measures ANOVA ▪ ANOVAs involving repeated measures/ within groups/subjects ▪ One independent variable with multiple levels tested within one subject group (e.g. drug A vs. drug B vs. placebo tested within same individuals at different times) ▪ ↓ variation effect between sample groups OSMOSIS.ORG 65
Figure 10.1 Examples demonstrating a one-way, two-way, and repeated measures ANOVA. The one-way ANOVA has one independent variable (medication type) with multiple levels (medications A, B, and C). The two-way ANOVA looks at two independent variables (medication type and age category) that each have multiple groups (medications A, B, and C; younger and older). The repeated measures ANOVA follows the same group of people over a period of time to measure the effects of the same medication over time. In this case, the independent variable is time, divided into three groups (one month, three months, and six months), and the dependent variable is systolic blood pressure. 66 OSMOSIS.ORG
Chapter 10 Biostatistics & Epidemiology: Parametric Tests Figure 10.2 All ANOVA tests assume that the groups have equal variance. A large variance means that the numbers are very spread out from the mean; a small variance means that the numbers are very close to the mean. Variances between groups are considered unequal when the variance of one group is greater than twice the variance of the other group. CORRELATION osms.it/correlation ▪ Investigates relationships between variables; determines strength, type (positive/negative) relationship ▪ Correlation coefﬁcient: r ( –1 > r < +1) ▫ Perfect positive correlation: r = +1 ▫ Perfect negative correlation: r = –1 ▫ No correlation: r = 0 ▫ Strong correlation: r > 0.5 < –0.5 ▫ Weak correlation: 0 < r < 0.5, or 0 > r > –0.5 ▪ Pearson product-moment coefﬁcient: interval/ratio data; calculates linear relationship degree between two variables ▪ Conﬁdence interval (CI): population based on correlation coefﬁcient ▫ Indicates range within population correlation coefﬁcient lies ▪ P-value for correlation coefﬁcient based on null hypothesis ▫ I.e. if true (p > 0.05), no correlation between variables ▪ Coefﬁcient of determination: r2 or R2 (0 < R2 < 1) ▫ Fraction of variation of variable of interest (x axis) due to another variable of interest (y axis) ▫ Remaining proportion due to natural variability ▫ Low R2 may indicate poor linear relationship, may be strong nonlinear relationship ▪ Eta-squared (η2): analogous to R2 for ANOVA ▪ Correlation ≠ causation, consider ▫ How strong is association? ▫ Does effect always follow cause? ▫ Is there a dose response? ▫ Relationship biologically plausible, coherent? ▫ Consistent ﬁnding? ▫ Other factors involved? ▫ Good experimental evidence? ▫ Analogous examples? OSMOSIS.ORG 67
Figure 10.3 Scatterplots are used to plot measurements, with one measured variable on each axis. Each data point represents one individual. A trend line is drawn to best represent the collection of data points on the plot, with roughly half the points above the line and the other half below the line. A perfect positive or negative correlation means that the trend line passes through every single data point. HYPOTHESIS TESTING osms.it/hypothesis-testing ▪ Calculating sample size required to test hypothesis ▪ Equations used for calculating power can also be used to calculate sample size for a predeﬁned alpha (0.05) ▪ Requires knowledge of ▫ Clinically important effect size (larger sample size needed to detect smaller effects) ▫ Surrogate endpoint use rather than direct outcome ▫ Desired power; alpha (if not 0.05); conﬁdence interval ▫ Statistical tests to be used ▫ Data lost to follow-up ▫ Test group SD; population of interest expected frequency within test group ▪ Statistician’s advice ▫ Optimize sample size, avoid underpowered studies, enable valid data interpretation LINEAR REGRESSION osms.it/linear-regression ▪ Simple linear regression: assumes linear relationship; slope ≠ 0; data points close to line ▪ Examine weight of two variables’ (x, y) effects; predict effects of x on y ▪ Fit best straight line to x, y plot of data ▫ Equation: y = bx + a (x and y are independent variables; b = slope of line (regression coefﬁcient); a = intercept ) ▪ 95% CI for slope range; larger sample → narrower CI; if range does not include zero → real correlation suggested 68 OSMOSIS.ORG ▪ p-value for null hypothesis ▫ No linear correlation (i.e. slope = 0; p < 0.05 → real correlation suggested) OTHER REGRESSION ANALYSES ▪ Multiple linear regression ▫ Examines effects of more than one variable on y ▪ Multiple nonlinear regression ▫ Examines correlations among nonlinear data, more than one independent variable
Chapter 10 Biostatistics & Epidemiology: Parametric Tests ▪ Logistic regression ▫ Predicts likelihood of categorical event in presence of multiple independent variables LOGISTIC REGRESSION osms.it/logistic-regression ▪ Predictive analysis: describes relationship between binary dependent variable (i.e. takes one of two values), multiple independent variables ▪ Assumptions ▫ Dichotomous outcome (e.g. yes/no; present/absent; dead/alive) ▫ No outliers: assess using z scores ▫ No intercorrelations: assess using correlation matrix ▪ May use logit (assumes log distribution of event’s probability)/probit (model assumes normal distribution) ▪ Rule of 10: stable values if based on minimum of 10 observations per independent variable ▪ Regression coefﬁcients: indicate contribution of individual independent variables; odds ratios ▪ Tests to assess signiﬁcance of independent variable ▫ Likelihood ratio test; Wald test ▪ Bayesian inference: prior (known) distributions for regression coefﬁcients; conjugate prior; automatic software (e.g. OpenBUGS, JAGS to simulate priors) TYPE I & TYPE II ERRORS osms.it/type-I-and-type-II-errors POWER ▪ Refers to test probability correctly rejecting false null hypothesis ▪ Power: (1 – beta) ▫ Likelihood that statistically nonsigniﬁcant result is correct (i.e. not false negative—type II error) ▪ Medical research ▫ Power typically set at 0.80 ▪ Increasing power ▫ ↓ type II error chance; ↑ type I error chance ▪ Power increases when ↑ sample size, ↓ SD, ↑ effect size EFFECT SIZE ▪ Relationship strength between variables ▪ Statistical signiﬁcance does not necessarily indicate clinical signiﬁcance ▪ Random variation (SD) may ↓ differences between outcomes of interest between hypothesis’ test groups ▪ ES = X1 − X 2 SD ▫ ES is effect size ▫ X1 is the mean for Group 1 ▫ X2 is the mean for Group 2 ▫ SD is the standard deviation from either group OSMOSIS.ORG 69
▪ Adjust for variation in test groups with Cohen’s d (assumes each group’s SD is same) ▫ Cohen’s d = (mean 1 – mean 2)/SD ▫ 0.2 = small effect size ▫ 0.5 = medium effect size ▫ > 0.8 = large effect size SAMPLE SIZE ▪ Smaller sample size ▫ ↑ sampling error chance ▫ Lower power ▫ ↑ type II error chance (false negative) BAYESIAN THINKING ▪ Relates p-value to context ▫ Can involve complex mathematics ▪ Measures event probability given incomplete information ▪ Joint distribution between given information (usually probability density), experimental results 70 OSMOSIS.ORG Figure 10.4 A Type I error occurs when no true relationship exists between two variables, but the study concludes there is one; a type II error occurs when there is a true relationship between two variables, but the study concludes there is no relationship.