Confidence Intervals For Treatment Effect Research Articles

Rivaroxaban versus aspirin for prevention of covert brain infarcts in patients with embolic stroke of undetermined source: NAVIGATE ESUS MRI substudy.

BackgroundCovert brain infarcts are associated with important neurological morbidity. Their incidence in patients with embolic stroke of undetermined source (ESUS) is unknown.AimsTo assess the incidence of covert brain infarcts and cerebral microbleeds using MRI in a prospective substudy of the NAVIGATE ESUS randomized trial and to evaluate the effects of antithrombotic therapies.MethodsAt 87 sites in 15 countries, substudy participants were randomly assigned to receive rivaroxaban 15 mg daily or aspirin 100 mg daily and underwent brain MRI near randomization and after study termination. The primary outcome was incident brain infarct (clinical ischemic stroke or covert brain infarct). Brain infarcts and microbleeds were ascertained centrally by readers unaware of treatment. Treatment effects were estimated using logistic regression.ResultsAmong the 718 substudy participants with interpretable, paired MRIs, the mean age was 67 years and 61% were men with a median of 52 days between the qualifying ischemic stroke and randomization and a median of seven days between randomization and baseline MRI. During the median (IQR) 11 (12) month interval between scans, clinical ischemic strokes occurred in 27 (4%) participants, while 60 (9%) of the remaining participants had an incident covert brain infarct detected by MRI. Assignment to rivaroxaban was not associated with reduction in the incidence of brain infarct (OR 0.77, 95% CI 0.49, 1.2) or of covert brain infarct among those without clinical stroke (OR 0.85, 95% CI 0.50, 1.4). New microbleeds were observed in 7% and did not differ among those assigned rivaroxaban vs. aspirin (HR 0.95, 95% CI 0.52–1.7).ConclusionsIncident covert brain infarcts occurred in twice as many ESUS patients as a clinical ischemic stroke. Treatment with rivaroxaban compared with aspirin did not significantly reduce the incidence of covert brain infarcts or increase the incidence of microbleeds, but the confidence intervals for treatment effects were wide.Registration: https://www.clinicaltrials.gov. Unique identifier: NCT 02313909

New Approaches for Estimation of Effect Sizes and their Confidence Intervals for Treatment Effects from Randomized Controlled Trials.

Although Cohen's d and the growth modeling analysis (GMA) d from linear models are common standardized effect sizes used to convey treatment effects, popular statistical software packages do not include them in their standard outputs. This article demonstrated the use of statistical software with user-prescribed parameter functions (e.g., Mplus) to produce d for treatment effects from both classical analysis and GMA--along with their associated standard errors (SEs) and confidence intervals (CIs). A Monte Carlo study was conducted to examine bias in the SE and CI for GMA d obtained with Mplus and found that both estimates were more accurate when calculated by the software with the standard bootstrap than with the delta method, but the delta method estimates were less biased than respective estimates from extant post hoc equations. Thus, users of many statistical software packages (including SAS, R, and LISREL) should obtain d or GMA d and associated CIs directly. Researchers employing less versatile software--and meta-analysts including ds and GMA ds in their syntheses of treatment effects--should continue to use the conventional post hoc equations. Biases in SEs and CIs for effect sizes obtained with them are ignorable and point estimates of d and GMA d are the same whether obtained directly from the software or with post hoc equations.

Determining postoperative outcomes after cleft palate repair: A systematic review and meta-analysis

BackgroundA lack of high-level evidence exists on the outcomes of different cleft palate repair techniques. A critical appreciation for the complication rates of common repair techniques is paramount to optimize cleft palate care. MethodsA literature search was conducted for articles on the measurement of fistula and velopharyngeal insufficiency (VPI) rates following cleft palate repair. Study quality was determined using validated scales. The heterogeneity between studies was evaluated using the I2 statistic. Random-effect model analysis and forest plots were used to report pooled relative risks (RRs) with 95% confidence intervals for treatment effect. P-values of 0.05 were considered statistically significant. ResultsOf 2386 studies retrieved, 852 underwent screening and 227 met inclusion criteria (130 studies (57%) on fistulas and 122 studies (54%) on VPI). Meta-analyses were performed using 32 studies. The Furlow technique was associated with less postoperative fistulae than the von Langenbeck and Veau/Wardill/Kilner techniques (RR = 0.56 [0.39–0.79], p < 0.01 and RR = 0.25 [0.12–0.52], p < 0.01, respectively). One-stage repair was associated with less fistulae compared to two-stage repair (RR = 0.42 [0.19–0.96], p = 0.04). The Furlow repair was also associated with a less VPI than the Bardach palatoplasty (RR = 0.41 [0.23, 0.71], p < 0.01), and the one-stage repair was associated with a reduction in VPI rates compared to two-stage repair (RR = 0.55 [0.32, 0.95], p = 0.03). ConclusionThe Furlow repair is associated with less risk of fistula formation than the von Langenbeck and Veau/Wardill/Kilner techniques and less VPI compared to the Bardach repair. One-stage repair is associated with less risk of fistula formation and VPI than two-stage repair.

Slower Progress of Aortic Valve Calcification With Vitamin K Supplementation

Calcific aortic stenosis is a common degenerative disease characterized by progressive aortic valve calcification (AVC).1 Effective medical treatment options to retard the progression of AVC are sparse.1 Epidemiological data point to vitamin K as a potential protective factor for cardiovascular health, particularly for protection against vascular calcification.2,3 Matrix Gla-protein (MGP), a potent inhibitor of cardiovascular calcification, requires vitamin K for posttranslational carboxylation and hence full bioactivity.4 Thus, vitamin K supplementation might retard the progression of AVC.1,2 Dephosphorylated undercarboxylated MGP (dp-ucMGP) serves as a circulating marker for vitamin K deficiency.2,3 We performed a 12-month prospective, single-center, open-label, randomized interventional trial in patients with asymptomatic or mildly symptomatic AVC. Written informed consent was obtained before inclusion in the trial (URL: http://www.clinicaltrials.gov. Unique identifier: NCT00785109; RWTH Aachen Institutional Review Board No. 165/08). Inclusion criterion was a peak flow velocity exceeding 2 m/s. The main exclusion criteria were chronic kidney disease (estimated glomerular filtration rate <60 mL·min−1·1.73 m−2), expected valve replacement within the next year, and anticoagulation with vitamin K antagonists. Patients were randomized 1:1 to receive 2 mg phytomenadione (vitamin …

A "placement of death" approach for studies of treatment effects on ICU length of stay.

Length of stay in the intensive care unit (ICU) is a common outcome measure in randomized trials of ICU interventions. Because many patients die in the ICU, it is difficult to disentangle treatment effects on length of stay from effects on mortality; conventional analyses depend on assumptions that are often unstated and hard to interpret or check. We adapt a proposal from Rosenbaum that addresses concerns about selection bias and makes its assumptions explicit. A composite outcome is constructed that equals ICU length of stay if the patient was discharged alive and indicates death otherwise. Given any preference ordering that compares death with possible lengths of stay, we can estimate the intervention's effects on the composite outcome distribution. Sensitivity analyses can show results for different preference orderings. We discuss methods for constructing approximate confidence intervals for treatment effects on quantiles of the outcome distribution or on proportions of patients with outcomes preferable to various cutoffs. Strengths and weaknesses of possible primary significance tests (including the Wilcoxon-Mann-Whitney rank sum test and a heteroskedasticity-robust variant due to Brunner and Munzel) are reviewed. An illustrative example reanalyzes a randomized trial of an ICU staffing intervention.

Recursive partitioning for heterogeneous causal effects

In this paper we propose methods for estimating heterogeneity in causal effects in experimental and observational studies and for conducting hypothesis tests about the magnitude of differences in treatment effects across subsets of the population. We provide a data-driven approach to partition the data into subpopulations that differ in the magnitude of their treatment effects. The approach enables the construction of valid confidence intervals for treatment effects, even with many covariates relative to the sample size, and without “sparsity” assumptions. We propose an “honest” approach to estimation, whereby one sample is used to construct the partition and another to estimate treatment effects for each subpopulation. Our approach builds on regression tree methods, modified to optimize for goodness of fit in treatment effects and to account for honest estimation. Our model selection criterion anticipates that bias will be eliminated by honest estimation and also accounts for the effect of making additional splits on the variance of treatment effect estimates within each subpopulation. We address the challenge that the “ground truth” for a causal effect is not observed for any individual unit, so that standard approaches to cross-validation must be modified. Through a simulation study, we show that for our preferred method honest estimation results in nominal coverage for 90% confidence intervals, whereas coverage ranges between 74% and 84% for nonhonest approaches. Honest estimation requires estimating the model with a smaller sample size; the cost in terms of mean squared error of treatment effects for our preferred method ranges between 7–22%.

Robust Nonparametric Confidence Intervals for Regression-Discontinuity Designs

In the regression-discontinuity (RD) design, units are assigned to treatment based on whether their value of an observed covariate exceeds a known cutoff. In this design, local polynomial estimators are now routinely employed to construct confidence intervals for treatment effects. The performance of these confidence intervals in applications, however, may be seriously hampered by their sensitivity to the specific bandwidth employed. Available bandwidth selectors typically yield a “large” bandwidth, leading to data-driven confidence intervals that may be biased, with empirical coverage well below their nominal target. We propose new theory-based, more robust confidence interval estimators for average treatment effects at the cutoff in sharp RD, sharp kink RD, fuzzy RD, and fuzzy kink RD designs. Our proposed confidence intervals are constructed using a bias-corrected RD estimator together with a novel standard error estimator. For practical implementation, we discuss mean squared error optimal bandwidths, which are by construction not valid for conventional confidence intervals but are valid with our robust approach, and consistent standard error estimators based on our new variance formulas. In a special case of practical interest, our procedure amounts to running a quadratic instead of a linear local regression. More generally, our results give a formal justification to simple inference procedures based on increasing the order of the local polynomial estimator employed. We find in a simulation study that our confidence intervals exhibit close-to-correct empirical coverage and good empirical interval length on average, remarkably improving upon the alternatives available in the literature. All results are readily available in R and STATA using our companion software packages described in Calonico, Cattaneo, and Titiunik (2014d, 2014b).

Approaches to inverse-probability-of-treatment–weighted estimation with concurrent treatments

ObjectivesIn a setting with two concurrent treatments, inverse-probability-of-treatment weights can be used to estimate the joint treatment effects or the marginal effect of one treatment while taking the other to be a confounder. We explore these two approaches in a study of intravenous iron use in hemodialysis patients treated concurrently with epoetin alfa (EPO). Study Design and SettingWe linked US Renal Data System data with electronic health records (2004–2008) from a large dialysis provider. Using a retrospective cohort design with 776,203 records from 117,050 regular hemodialysis patients, we examined a composite outcome: mortality, myocardial infarction, or stroke. ResultsWith EPO as a joint treatment, inverse-probability-of-treatment weights were unstable, confidence intervals for treatment effects were wide, covariate balance was unsatisfactory, and the treatment and outcome models were sensitive to omission of the baseline EPO covariate. By handling EPO exposure as a confounder instead of a joint treatment, we derived stable weights and balanced treatment groups on measured covariates. ConclusionsIn settings with concurrent treatments, if only one treatment is of interest, then including the other in the treatment model as a confounder may result in more stable treatment effect estimates. Otherwise, extreme weights may necessitate additional analysis steps.

A Predictive Bayesian Approach to the Design and Analysis of Bridging Studies

Pharmaceutical product development culminates in confirmatory trials whose evidence for the product's efficacy and safety supports regulatory approval for marketing. Regulatory agencies in countries whose patients were not included in the confirmatory trials often require confirmation of efficacy and safety in their patient populations, which may be accomplished by carrying out bridging studies to establish consistency for local patients of the effects demonstrated by the original trials. This article describes and illustrates an approach for designing and analyzing bridging studies that fully incorporates the information provided by the original trials. The approach determines probability contours or regions of joint predictive intervals for treatment effect and response variability, or endpoints of treatment effect confidence intervals, that are functions of the findings from the original trials, the sample sizes for the bridging studies, and possible deviations from complete consistency with the original trials. The bridging studies are judged consistent with the original trials if their findings fall within the probability contours or regions. Regulatory considerations determine the region definitions and appropriate probability levels. Producer and consumer risks provide a way to assess alternative region and probability choices. [Supplemental materials are available for this article. Go to the Publisher's online edition of the Journal of Biopharmaceutical Statistics for the following free supplemental resource: Appendix 2: R code for Calculations.]

Confidence intervals for treatment effect from restricted maximum likelihood

An explicit form of confidence intervals for the treatment effect in random effects meta-analysis model obtained from Harville–Jeske–Kenward–Roger approach is given. These restricted likelihood based intervals are compared to alternative procedures commonly used in collaborative studies when the number of participants is small and study-specific variances are heterogeneous. Monte Carlo simulation experiments show that the former intervals have quite conservative coverage probabilities and favor the latter intervals.

The weighted log-rank class under truncated binomial design: saddlepoint p-values and confidence intervals

The randomization design used to collect the data provides basis for the exact distributions of the permutation tests. The truncated binomial design is one of the commonly used designs for forcing balance in clinical trials to eliminate experimental bias. In this article, we consider the exact distribution of the weighted log-rank class of tests for censored data under the truncated binomial design. A double saddlepoint approximation for p-values of this class is derived under the truncated binomial design. The speed and accuracy of the saddlepoint approximation over the normal asymptotic facilitate the inversion of the weighted log-rank tests to determine nominal 95% confidence intervals for treatment effect with right censored data.

568 A permutation-based confidence interval for treatment effect in the identified subset of sensitive patients in biomarker-adaptive threshold design

Indoleamine 2,3-dioxygenase (IDO) is a tryptophan-catabolizing enzyme that induces immune tolerance. The purpose of the present study is to investigate IDO expression and its functional role in ovarian cancer cells in vitro and in vivo.IDO expression was immunohistochemically scored in surgically-resected ovarian cancer tissues (n = 60), and its association with tumor-infiltrating lymphocyte (TIL) count or patient survival was analyzed. Next, IDO cDNA was transfected into the human ovarian carcinoma cell line SKOV3, establishing stable clones of IDO-overexpressing cells (SK-IDO). SK-IDO cells were characterized in vitro as well as in vivo using a nude mouse xenograft model.High IDO expression in tumor cells was found in 34 (56.7%) cases and was correlated with a reduced number of CD8+ TIL. Patients with high IDO expression had significantly impaired overall and progression-free survival compared to patients with no or low IDO expression. There were no significant differences in in vitro cell proliferation, migration, invasion, or chemosensitivity to paclitaxel between the SK-IDO and control vector-transfected (SK-pcDNA) cells. However, tumor peritoneal dissemination was significantly increased in SK-IDO-xenografted mice compared to SK-pcDNA-xenografted mice. This tumor-progressive effect in SK-IDO-xenografted mice was abrogated by oral administration of the IDO inhibitor 1-methyl-tryptophan (1-MT). Finally, treatment with weekly i.p. paclitaxel combined with daily administration of 1-MT significantly prolonged the survival of the SK-IDO-xenografted mice compared to treatment with paclitaxel alone.These results suggest that IDO is involved in ovarian cancer progression in vivo and may be a promising therapeutic target for advanced ovarian cancer.

Nonparametric confidence intervals for Tmax in sequence‐stratified crossover studies

Tmax is the time associated with the maximum serum or plasma drug concentration achieved following a dose. While Tmax is continuous in theory, it is usually discrete in practice because it is equated to a nominal sampling time in the noncompartmental pharmacokinetics approach. For a 2-treatment crossover design, a Hodges-Lehmann method exists for a confidence interval on treatment differences. For appropriately designed crossover studies with more than two treatments, a new median-scaling method is proposed to obtain estimates and confidence intervals for treatment effects. A simulation study was done comparing this new method with two previously described rank-based nonparametric methods, a stratified ranks method and a signed ranks method due to Ohrvik. The Normal theory, a nonparametric confidence interval approach without adjustment for periods, and a nonparametric bootstrap method were also compared. Results show that less dense sampling and period effects cause increases in confidence interval length. The Normal theory method can be liberal (i.e. less than nominal coverage) if there is a true treatment effect. The nonparametric methods tend to be conservative with regard to coverage probability and among them the median-scaling method is least conservative and has shortest confidence intervals. The stratified ranks method was the most conservative and had very long confidence intervals. The bootstrap method was generally less conservative than the median-scaling method, but it tended to have longer confidence intervals. Overall, the median-scaling method had the best combination of coverage and confidence interval length. All methods performed adequately with respect to bias.

Quantifying effects in two-sample environmental experiments using bootstrap confidence intervals

Two-sample experiments (paired or unpaired) are often used to analyze treatment effects in life and environmental sciences. Quantifying an effect can be achieved by estimating the difference in center of location between a treated and a control sample. In unpaired experiments, a shift in scale is also of interest. Non-normal data distributions can thereby impose a serious challenge for obtaining accurate confidence intervals for treatment effects. To study the effects of non-normality we analyzed robust and non-robust measures of treatment effects: differences of averages, medians, standard deviations, and normalized median absolute deviations in case of unpaired experiments, and average of differences and median of differences in case of paired experiments. A Monte Carlo study using bivariate lognormal distributions was carried out to evaluate coverage performances and lengths of four types of nonparametric bootstrap confidence intervals, namely normal, Student's t, percentile, and BCa for the estimated measures. The robust measures produced smaller coverage errors than their non-robust counterparts. On the other hand, the robust versions gave average confidence interval lengths approximately 1.5 times larger. In unpaired experiments, BCa confidence intervals performed best, while in paired experiments, Student's t was as good as BCa intervals. Monte Carlo results are discussed and recommendations on data sizes are presented. In an application to physiological source–sink manipulation experiments with sunflower, we quantify the effect of an increased or decreased source–sink ratio on the percentage of unfilled grains and the dry mass of a grain. In an application to laboratory experiments with wastewater, we quantify the disinfection effect of predatory microorganisms. The presented bootstrap method to compare two samples is broadly applicable to measured or modeled data from the entire range of environmental research and beyond.

Galantamine for Alzheimer's disease and mild cognitive impairment

Galantamine is a specific, competitive, and reversible acetylcholinesterase inhibitor. To assess the clinical effects of galantamine in patients with mild cognitive impairment (MCI), probable or possible Alzheimer's disease (AD), and potential moderators of effect. The trials were identified from a search of the Specialized Register of the Cochrane Dementia and Cognitive Improvement Group, last updated on 25 April 2005 using the terms galanthamin*, galantamin* and Reminyl. Published reviews were inspected for further sources. Additional information was collected from unpublished clinical research reports for galantamine obtained from Janssen and from http://www.clinicalstudyresults.org/. Trials selected were randomised, double-blind, parallel-group comparisons of galantamine with placebo for a treatment duration of greater than 4 weeks in subjects with MCI or AD. Data were extracted independently by the reviewers and pooled where appropriate and possible. Outcomes of interest include the clinical global impression of change (CIBIC-plus or CGIC), Alzheimer's Disease Assessment Scale-cognitive sub scale (ADAS-cog), Alzheimer's Disease Cooperative Study/Activities of Daily Living (ADCS-ADL), Disability Assessment for Dementia scale (DAD) and Neuropsychiatric Inventory (NPI). Potential moderating variables of treatment effect assessed included trial duration, dose, and diagnosis of possible versus probable Alzheimer's disease. Ten trials with a total 6805 subjects were included in the analysis. Treatment with galantamine led to a significantly greater proportion of subjects with improved or unchanged global rating scale rating (k = 8 studies), at all dosing levels except for 8 mg/d . Confidence intervals for the ORs overlapped across the dose range of 16 mg to 36 mg per day, with point estimates of 1.6 - 1.8 when analysed with the intention-to-treat sample. Treatment with galantamine also led to significantly greater reduction in ADAS-cog score at all dosing levels (k = 8), with greater effect over six months compared to three months. Confidence intervals again overlapped. Point estimate of effect was lower for 8 mg/d but similar for 16 mg to 36 mg per day. For example, treatment effect for 24 mg/d over six months was 3.1 point reduction in ADAS-cog (95%CI 2.6-3.7, k = 4, ITT).ADCS-ADL, DAD and NPI were reported only in a small proportion of trials: all showed significant treatment effect in some individual trials at least. Confidence interval of treatment effect for the one trial recruiting patients with possible AD overlapped with the other seven recruiting patients with probable AD. Galantamine's adverse effects appeared similar to those of other cholinesterase inhibitors and to be dose related. Prolong release / once daily formulation of galantamine at 16 - 24mg/d was found to have similar efficacy and side-effect profile as the equivalent twice-daily regime. Data from the two MCI trials suggest marginal clinical benefit, but a yet unexplained excess in death rate. Subjects in these trials were similar to those seen in earlier anti dementia AD trials, consisting primarily of mildly to moderately impaired outpatients. Galantamine's effect on more severely impaired subjects has not yet been assessed.Nevertheless, this review shows consistent positive effects for galantamine for trials of three to six months' duration. Although there was not a statistically significant dose-response effect, doses above 8 mg/d were, for the most part, consistently statistically significant. Galantamine's safety profile in AD is similar to that of other cholinesterase inhibitors with respect to cholinergically mediated gastrointestinal symptoms. It appears that doses of 16 mg/d were best tolerated in the single trial where medication was titrated over a four week period, and because this dose showed statistically indistinguishable efficacy with higher doses, it is probably most preferable initially. Longer term use of galantamine has not been assessed in a controlled fashion. Galantamine use in MCI is not recommended due to its association with an excess death rate.

Randomization inference for balanced cluster-randomized trials

This paper discusses the choice of randomization tests for inferences from cluster-randomized trials that have been designed to ensure a balanced allocation of clusters to treatments. Methods for covariate-adjusted randomization tests are reviewed and their application to balanced cluster-randomized trials discussed. Two cluster-randomized trials with balanced designs are used to illustrate the choices that can be made in selecting a randomization test, and methods for obtaining confidence intervals for treatment effects are illustrated. The balance imposed by the randomization in these trials makes adjustment for covariates less beneficial than for an unbalanced design. However, the adjusted analyses do not appear generally to have worse properties than the unadjusted ones, and may provide protection against any imbalance that has not been controlled for in the design. The only case when adjustment for covariates may result in worse precision is when a large number of cluster-level covariates are included in the analysis. An expression is provided that allows the size of this effect to be calculated for any given set of cluster-level covariates.

Resampling methods for confidence intervals in group sequential trials

SUMMARY Resampling methods are introduced for the construction of confidence intervals for treatment effects associated with the primary and secondary endpoints of a clinical trial whose stopping rule is specified by a group sequential test. These methods are nonparametric and compare favourably with the exact methods that assume the responses to be normally distributed.

Applied Multivariate Statistical Analysis.

(NOTE: Each chapter begins with an Introduction, and concludes with Exercises and References.) I. GETTING STARTED. 1. Aspects of Multivariate Analysis. Applications of Multivariate Techniques. The Organization of Data. Data Displays and Pictorial Representations. Distance. Final Comments. 2. Matrix Algebra and Random Vectors. Some Basics of Matrix and Vector Algebra. Positive Definite Matrices. A Square-Root Matrix. Random Vectors and Matrices. Mean Vectors and Covariance Matrices. Matrix Inequalities and Maximization. Supplement 2A Vectors and Matrices: Basic Concepts. 3. Sample Geometry and Random Sampling. The Geometry of the Sample. Random Samples and the Expected Values of the Sample Mean and Covariance Matrix. Generalized Variance. Sample Mean, Covariance, and Correlation as Matrix Operations. Sample Values of Linear Combinations of Variables. 4. The Multivariate Normal Distribution. The Multivariate Normal Density and Its Properties. Sampling from a Multivariate Normal Distribution and Maximum Likelihood Estimation. The Sampling Distribution of 'X and S. Large-Sample Behavior of 'X and S. Assessing the Assumption of Normality. Detecting Outliners and Data Cleaning. Transformations to Near Normality. II. INFERENCES ABOUT MULTIVARIATE MEANS AND LINEAR MODELS. 5. Inferences About a Mean Vector. The Plausibility of ...m0 as a Value for a Normal Population Mean. Hotelling's T 2 and Likelihood Ratio Tests. Confidence Regions and Simultaneous Comparisons of Component Means. Large Sample Inferences about a Population Mean Vector. Multivariate Quality Control Charts. Inferences about Mean Vectors When Some Observations Are Missing. Difficulties Due To Time Dependence in Multivariate Observations. Supplement 5A Simultaneous Confidence Intervals and Ellipses as Shadows of the p-Dimensional Ellipsoids. 6. Comparisons of Several Multivariate Means. Paired Comparisons and a Repeated Measures Design. Comparing Mean Vectors from Two Populations. Comparison of Several Multivariate Population Means (One-Way MANOVA). Simultaneous Confidence Intervals for Treatment Effects. Two-Way Multivariate Analysis of Variance. Profile Analysis. Repealed Measures, Designs, and Growth Curves. Perspectives and a Strategy for Analyzing Multivariate Models. 7. Multivariate Linear Regression Models. The Classical Linear Regression Model. Least Squares Estimation. Inferences About the Regression Model. Inferences from the Estimated Regression Function. Model Checking and Other Aspects of Regression. Multivariate Multiple Regression. The Concept of Linear Regression. Comparing the Two Formulations of the Regression Model. Multiple Regression Models with Time Dependant Errors. Supplement 7A The Distribution of the Likelihood Ratio for the Multivariate Regression Model. III. ANALYSIS OF A COVARIANCE STRUCTURE. 8. Principal Components. Population Principal Components. Summarizing Sample Variation by Principal Components. Graphing the Principal Components. Large-Sample Inferences. Monitoring Quality with Principal Components. Supplement 8A The Geometry of the Sample Principal Component Approximation. 9. Factor Analysis and Inference for Structured Covariance Matrices. The Orthogonal Factor Model. Methods of Estimation. Factor Rotation. Factor Scores. Perspectives and a Strategy for Factor Analysis. Structural Equation Models. Supplement 9A Some Computational Details for Maximum Likelihood Estimation. 10. Canonical Correlation Analysis Canonical Variates and Canonical Correlations. Interpreting the Population Canonical Variables. The Sample Canonical Variates and Sample Canonical Correlations. Additional Sample Descriptive Measures. Large Sample Inferences. IV. CLASSIFICATION AND GROUPING TECHNIQUES. 11. Discrimination and Classification. Separation and Classification for Two Populations. Classifications with Two Multivariate Normal Populations. Evaluating Classification Functions. Fisher's Discriminant Function...nSeparation of Populations. Classification with Several Populations. Fisher's Method for Discriminating among Several Populations. Final Comments. 12. Clustering, Distance Methods and Ordination. Similarity Measures. Hierarchical Clustering Methods. Nonhierarchical Clustering Methods. Multidimensional Scaling. Correspondence Analysis. Biplots for Viewing Sample Units and Variables. Procustes Analysis: A Method for Comparing Configurations. Appendix. Standard Normal Probabilities. Student's t-Distribution Percentage Points. ...c2 Distribution Percentage Points. F-Distribution Percentage Points. F-Distribution Percentage Points (...a = .10). F-Distribution Percentage Points (...a = .05). F-Distribution Percentage Points (...a = .01). Data Index. Subject Index.