The EBM 3-step approach How an article should be appraised, in 3 steps: Step 1 – Are the results of the study (internally) valid? Step 2 – What are the results? Step 3 – How can I apply these results to patient care? Guyatt and Rennie, Users’ guide to the medical literature, 2002
The Cochrane Collaboration Risk of Bias Tool http://www.cochrane.org
The ultimate goal of any clinical or scientific observation is the appraisal of causality
Force:* precisely defined (p<0.05, weaker criterion) and with strong relative risk (≤0.83 or ≥1.20) in the absence of multiplicity issues (stronger criterion) Consistency:* results in favor of the association must be confirmed in other studies Temporality: exposition must precede in a realistic fashion the event Coherence: hypothetical cause-effect relationship is not in contrast with other biologic or natural history findings Mente et al, Arch Intern Med 2009 Bradford Hill causality criteria *statistics is important here
Biologic gradient:* exposition dose and risk of disease are positively (or negatively) associated on a continuum Experimental: experimental evidence from laboratory studies (weaker criterion) or randomized clinical trials (stronger criterion) Specificity: exposition is associated with a single disease (does not apply to multifactorial conditions) Plausibility: hypothetical cause-effect relationship makes sense from a biologic or clinical perspective (weaker criterion) Analogy: hypothetical cause-effect relationship is based on analogic reasoning (weaker criterion) Bradford Hill causality criteria Mente et al, Arch Intern Med 2009 *statistics is important here
Randomization Is the technique which defines experimental studies in humans (but not only in them), and enables the correct application of statistical tests of hypothesis in a frequentist framework (according to Ronal Fischer theory) Randomization means assigning at random a patient (or a study unit) to one of the treatments Over large numbers, randomization minimizes the risk of imbalances in patient or procedural features, but this does not hold true for small samples and for a large set of features
Any clinical or scientific comparison can be viewed as… A battle between an underlying hypothesis (null, H 0 ), stating that there is no meaningful difference or association (beyond random variability) between 2 or more populations of interest (from which we are sampling) and an alternative hypothesis (H 1 ), which implies that there is a non-random difference between such populations. Any statistical test is a test trying to convince us that H 0 is false (thus implying the working truthfulness of H 1 ).
Falsifiability Falsifiability or refutability of a statement, hypothesis, or theory is an inherent possibility to prove it to be false. A statement is called falsifiable if it is possible to conceive an observation or an argument which proves the statement in question to be false. In this sense, falsify is synonymous with nullify, meaning not "to commit fraud" but "show to be false
Statistical or clinical significance? Statistical and clinical significance are 2 very different concepts. A clinically significant difference, if demostrated beyond the play of chance, is clinically relevant and thus merits subsequent action (if costs and tolerability issues are not overcoming). A statistically significant difference is a probabilistic concept and should be viewed in light of the distance from the null hypothesis and the chosen significance threshold.
And this might be its universal population Samples and populations
But what if THIS is its universal population? Samples and populations
Any inference thus depend on our confidence in its likelihood Samples and populations
Alpha and type I error Whenever I perform a test, there is thus a risk of a FALSE POSITIVE result, ie REJECTING A TRUE null hypothesis. This error is called type I, is measured as alpha and its unit is the p value. The lower the p value, the lower the risk of falling into a type I error (ie the HIGHER the SPECIFICITY of the test).
Type I error is like a MIRAGE Because I see something that does NOT exist Alpha and type I error
Beta and type II error Whenever I perform a test, there is also a risk of a FALSE NEGATIVE result, ie NOT REJECTING A FALSE null hypothesis. This error is called type II, is measured as beta, and its unit is a probability. The complementary of beta is called power. The lower the beta, the lower the risk of missing a true difference (ie the HIGHER the SENSITIVITY of the test).
Type II error is like being BLIND Because I do NOT see something that exists Beta and type II error
Accuracy and precision Accuracy measures the distance from the true value distance Precision measures the spead in the measurements spread true value measurement
Accuracy and precision Thus: Precision expresses the extent of RANDOM ERROR Accuracy expresses the extent of SYSTEMATIC ERROR (ie bias)
Validity Internal validity entails both PRECISION and ACCURACY (ie does a study provide a truthful answer to the research question?) External validity expresses the extent to which the results can be applied to other contexts and settings. It corresponds to the distinction between SAMPLE and POPULATION)
Intention-to-treat analysis Intention-to-treat (ITT) analysis is an analysis based on the initial treatment intent, irrespectively of the treatment eventually administered. ITT analysis is intended to avoid various types of bias that can arise in intervention research, especially procedural, compliance and survivor bias. However, ITT dilutes the power to achieve statistically and clinically significant differences, especially as drop-in and drop-out rates rise.
Per-protocol analysis In contrast to the ITT analysis, the per-protocol (PP) analysis includes only those patients who complete the entire clinical trial or other particular procedure(s), or have complete data. In PP analysis each patient is categorized according to the actual treatment received, and not according to the originally intended treatment assignment. PP analysis is largely prone to bias, and is useful almost only in equivalence or non-inferiority studies.
ITT vs PP 100 pts enrolled RANDOMIZATION 50 pts to group A (more toxic) 50 pts to group B (conventional Rx, less toxic) 45 pts treated with A, 5 shifted to B because of poor global health (all 5 died) 50 patients treated with B (none died) ACTUAL THERAPY
ITT vs PP 100 pts enrolled RANDOMIZATION 50 pts to group A (more toxic) 50 pts to group B (conventional Rx, less toxic) 45 pts treated with A, 5 shifted to B because of poor global health (all 5 died) 50 patients treated with B (none died) ACTUAL THERAPY ITT p=0.021 in favor of B ITT: 10% mortality in group A vs 0% in group B, p=0.021 in favor of B
ITT vs PP 100 pts enrolled RANDOMIZATION 50 pts to group A (more toxic) 50 pts to group B (conventional Rx, less toxic) 45 pts treated with A, 5 shifted to B because of poor global health (all 5 died) 50 patients treated with B (none died) ACTUAL THERAPY ITT p=0.021 in favor of B ITT: 10% mortality in group A vs 0% in group B, p=0.021 in favor of B PP p=0.038 in favor of A PP: 0% (0/45) mortality in group A vs 9.1% (5/55) in group B, p=0.038 in favor of A
Characteristics: -summarises information well -discards a lot of information (dispersion??) Assumptions: -data are not skewed –distorts the mean –outliers make the mean very different -Measured on measurement scale –cannot find mean of a categorical measure ‘average’ stent diameter may be meaningless Mean (arithmetic)
What is it? –The one in the middle –Place values in order –Median is central Definition: –Equally distant from all other values Used for: –Ordinal data –Skewed data / outliers Median
Standard deviation (SD): –approximates population σ as N increases Advantages: –with mean enables powerful synthesis mean±1*SD 68% of data mean±2*SD 95% of data (1.96) mean±3*SD 99% of data (2.86) Disadvantages: –is based on normal assumptions 1 )( 2 - - N xx SD Standard deviation Variance
Interquartile range Variable type Continuous Patient IDLesion Length 1114 615 716 317 118 8 1019 921 1222 523 224 425 1327 16.5 23.5 25% to 75% percentile or 1° to 3° quartile Median 1st-3rd Quartile =16.5; 23.5 Interquartile Range =23.5-16.5=7.0
Coefficient of variation CV = x 100 Standard deviation Mean Coefficient of variation (CV) is a index of relative variability CV is dimensionless CV enables you to compare data dispersion of variables with different units of measurement
Point estimation & confidence intervals Using summary statistics (mean and standard deviation for normal variables, or proportion for categorical variable) and factoring sample size, we can build confidence intervals or test hypotheses that we are sampling from a given population or not This can be done by creating a powerful tool, which weighs our dispersion measures by means of the sample size: the standard error
First you need the SE We can easily build the standard error of a proportion, according to the following formula: Where variance=P*(1-P) and n is the sample size n SE = P * (1-P)
Point estimation & confidence intervals We can then create a simple test to check whether the summary estimate we have found can be compatible according to random variation with the corresponding reference population mean The Z test (when the population SD is known) and the t test (when the population SD is only estimated), are thus used, and both can be viewed as a signal to noise ratio
From the Z test… Signal to noise ratio Z score = Absolute difference in summary estimates = Signal Noise Standard error Results of z score correspond to a distinct tail probability of the Gaussian curve (eg 1.96 corresponds to a 0.025 one-tailed probability or 0.050 two-tailed probability)
SD …to confidence intervals Standard error (SE or SEM) can be used to test a hypothesis or create a confidence interval (CI) around a mean for a continuous variable (eg mortality rate) 95% CI = mean ± 2 SE n SE = 95% means that, if we repeat the study 20 times, 19 times out of 20 we will included the true population average
Ps and confidence intervals P values and confidence intervals are strictly connected Any hypothesis test providing a significant result (eg p=0.045) means that we can be confident at 95.5% that the population average difference lies far from zero (ie the null hypothesis)
significant difference (p<0.05) non significant difference (p>0.05) HoHo Important difference Trivial difference P values and confidence intervals
Power and sample size Whenever designing a study or analyzing a dataset, it is important to estimate the sample size or the power of the comparison. SAMPLE SIZE Setting a specific alpha and a specific beta, you calculate the necessary sample size given the average inter-group difference and its variation. POWER Given a specific sample size and alpha, in light of the calculated average inter-group difference and its variation, you obtain an estimate of the power (ie 1-beta).
Hierachy of analysis A statistical analysis: –Univariate (e.g. when describing mean or standar d deviation) –Bivariate (e.g. when comparing age in men adn women ) –Multivariable (e.g. when appraising how age and gender impact on the risk of death) –Multivariate (e.g. when appraising how age and gender simultaneously impact on risk of death and hospital costs)
Variables PAIRED OR REPEATED MEASURES UNPAIRED OR INDEPENDENT MEASURES eg blood pressure measured twice in the same patients at different times eg blood pressure measured in several different groups of patients only once Types of variables
Variables nominalordinaldiscretecontinuous orderedcategories ranks counting measuring Types of variablesQUANTITYCATEGORY
Categorical data: compare proportions in groups Continuous data: compare means or medians in groups Normal data; use t test Non-normal data; use Mann Whitney U test Normal data; use ANOVA Non-normal data; use Kruskal Wallis Are data categorical or continuous? How many groups? Two groups; normal data? More than two groups; normal data? Statistical tests
Student t test It is used to test the null hypothesis that the means of two normally distributed populations are equal Given two data sets (each with its mean, SD and number of data points) the t test determines whether the means are distinct, provided that the underlying distributions can be assumed to be normal the Student t test should be used if the variances (not known) of the two populations are also assumed to be equal; the form of the test used when this assumption is dropped is sometimes called Welch's t test
1-way ANOVA As with the t-test, ANOVA is appropriate when the data are continuous, when the groups are assumed to have similar variances, and when the data are normally distributed ANOVA is based upon a comparison of variance attributable to the independent variable (variability between groups or conditions) relative to the variance within groups resulting from random chance. In fact, the formula involves dividing the between-group variance estimate by the within-group variance estimate
Kruskal Wallis test Post-hoc analysis with Mann Withney U and Bonferroni correction
Three (or more) paired groups Again ask yourself… Parametric or not? If parametric: ANOVA for repeated measures in SPSS… in the General Linear Model If non-parametric: Friedman test Compare continuous variables
CAMELOT, JAMA 2004 A mixed-design ANOVA is used to test for differences between independent groups whilst subjecting participants to repeated measures. In a mixed-design ANOVA model, one factor is a between-subjects variable (drug) and the other is within-subjects variable (BP) 2-way ANOVA
Binomial test Is the percentage of diabetics in this sample comparable with the known chronic AF population? We assume the population rate is at 15%
Compare discrete variables The second basis is the “observed”-“expected” relation
Absolute Risk (AR) 7.9% (47/592) & 15.1% (89/591) Absolute Risk Reduction (ARR) 7.9% (47/592) – 15.1% (89/591) = -7.2% Relative Risk (RR) 7.9% (47/592) / 15.1% (89/591) = 0.52 (given an equivalence value of 1) Relative Risk Reduction (RRR) 1 – 0.52 = 0.48 or 48% Odds Ratio (OR) 8.6% (47/545) / 17.7% (89/502) = 0.49 (given an equivalence value of 1) Compare event rates
the chi-square test was used to determine differences between groups with respect to the primary and secondary end points. Odds ratios and their 95 percent confidence intervals were calculated. Comparisons of patient characteristics and survival outcomes were tested with the chi-square test, the chi-square test for trend, Fisher's exact test, or Student's t-test, as appropriate. This is a sub-group ! Bonferroni ! The level of significant p-value should be divided by the number of tests performed… Or the computed p-value, multiplied for the number of tests… P=0.12 and not P=0.04 !! Post-hoc groups Wenzel et al, NEJM 2004
ab cd Event No event ExpCtrl s1s1s1s1 s2s2s2s2 r2r2r2r2 r1r1r1r1 N P = s 1 ! * s 2 ! * r 1 ! * r 2 ! N! * a ! * b! * c! * d! Fisher Exact test
The McNemar test is a hypothesis test to compare categorical variables in related samples. For instance, how can I appraise the statistical significance of change into symptom status (asymptomatic vs symptomatic) in the same patients over time? The McNemar test exploits the discordant pairs to generate a p value McNemar test Follow-up symptomatic Follow-up asymptomatic Baseline symptomatic 153 Baseline asymptomatic 517 Follow-up symptomatic Follow-up asymptomatic Baseline symptomatic 150 Baseline asymptomatic 817 p=0.72 at McNemar testp=0.013 at McNemar test
Biostatistics is best seen as a set of different tools and methods which are used according the problem at hand. Nobody is experienced in statistics at beginning, and only by facing everyday real-world problems you can familiarize yourself with different techniques and approaches. In general terms, it is also crucial to remember that the easiest and simplest way to solve a statistical problem, if appropriate, is also the best and the one recommended by reviewers.
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Giuseppe Biondi-Zoccai, MD Sapienza University of Rome, Latina, Italy email@example.com@gmail.com Medical statistics for cardiovascular disease Part 2
It can be intuitively understood that it is the line that minimizes the differences between observed values (y i ) and estimated values (y i ’) Which of these different possible lines that I can graphically trace and compute is the best regression line? Linear regression
Correlation The square root of the coefficient of determination (R 2 ) is the correlation coefficient (R) and shows the degree of linear association between 2 continuous variables, but disregards causation. Assumes values between -1.0 (negative association), 0 (no association), and +1.0 (positive association). It can be summarized as a point summary estimate, with specific standard error, 95% confidence interval, and p value. K. Pearson
Dangers of not plotting data 4 sets of data: all with the same R=0.81!* *At linear regression analysis
What about non-linear associations? Each number correspond to the correlation coefficient for linear association (R)!!!
Pearson vs Spearman Whenever the independent and dependent variables can be assumed to belong to normal distributions, the Pearson linear correlation method can be used, maximizing statistical power and yield. Whenever the data are sparse, rare, and/or not belonging to normal distributions, the non- parametric Spearman correlation method should be used, which yields the rank correlation coefficient (rho), but not its R 2. C. Spearman
Bland Altman plot Mean of measurement A and B in each case Difference of A – B in each case
Regression to the mean: don’t bet on past rookies of the year!
Logistic regression We model ln [p/(1-p)] instead of just p, and the linear model is written : ln [p/(1-p)] = ln(p) – ln(1-p) = β 0 + β 1 *X Logistic regression is based on the logit which transforms a dichotomous dependent variable into a continuous one
Generalized Linear Models All generalized linear models have three components : – Random component identifies the response variable and assumes a probability distribution for it – Systematic component specifies the explanatory variables used as predictors in the model (linear predictor). – Link describes the functional relationship between the systematic component and the expected value (mean) of the random component. The GLM relates a function of that mean to the explanatory variables through a prediction equation having linear form. The model formula states that: g(µ) = α + β 1 x 1 + … + β k x k
Generalized Linear Models Through differing link functions, GLM corresponds to other well known models DistributionNameLink FunctionMean Function NormalIdentity Exponential Inverse Gamma Inverse Gaussian Inverse squared PoissonLog BinomialLogit
Patients experiencing one or more events are called responders Patients who, at the end of the observational period or before such time, get out of the study without having experienced any event, are called censored Survival analysis
ABCDEFABCDEF 2 4 6 8 10 12 x Withdrawn Study end Lost x ABCDEFABCDEF 2 4 6 8 10 12 x Withdrawn Study end Lost x Study end A and F: events B, C, D and E: censored
Independent and dependent variables Predictors Regressors Explanatory Prognostic Factors Manipulated Response IndependentDependent Ind. Var. : have an effect on... Dep. Var.: is influenced by … Goal is to explain the variation in the dependent variable by other variables simultaneously. Multivariable statistical methods
One Dep. Var. ~ One Ind. Var. D. V. Qualitative Quantitative I.V. Qualitative I.V. Simple RegressionAnova 1Logistic Reg.Chi² Bivariate statistical methods
One Dep. Var. ~ Several Ind. Var. D. V. Qualitative Quantitative I.V. Qualitative I.V. Simple RegressionAnova 1Logistic Reg.Chi² Multivariable statistical methods
Multivariable analysis The methods mentioned have specific application domains depending on the nature of the variables involved in the analysis. But conceptually and qua calculation there are a lot of similarities between these techniques. Each of the multivariable methods evaluates the effect of an independent variable on the dependent variable, controlling for the effect of other independent variables. Methods such as multiple regression, multi-factor ANOVA, analysis of covariance have the same assumptions towards the distribution of the dependent variable. We will learn more about the concepts of multivariable analysis by reviewing the simple linear regression model.
Multiple linear regression Simple linear regression is a statistical model to predict the value of one continuous variable Y (dependent, response) from another continuous variable X (independent, predictor, covariate, prognostic factor). Multiple linear regression is a natural extension of the simple linear regression model – We use it to investigate the effect on the response variable of several predictor variables, simultaneously – It is a hypothetical model of the relationship between several independent variables and a response variable. Let’s start by reviewing the concepts of the simple linear regression model.
Multiple regression models Model terms may be divided into the following categories – Constant term – Linear terms / main effects (e.g. X 1 ) – Interaction terms (e.g. X 1 X 2 ) – Quadratic terms (e.g. X 1 2 ) – Cubic terms (e.g. X 1 3 ) Models are usually described by the highest term present – Linear models have only linear terms – Interaction models have linear and interaction terms – Quadratic models have linear, quadratic and first order interaction terms – Cubic models have terms up to third order.
The model-building process Source: Applied Linear Statistical Models, Neter, Kutner, Nachtsheim, Wasserman
AIC and BIC AIC (Akaike Information Criterion) and BIC (Swarz Information Criterion) are two popular model selection methods. They not only reward goodness of fit, but also include a penalty that is an increasing function of the number of estimated parameters. This penalty discourages overfitting. The preferred model is the one with the lowest value for AIC or for BIC. These criteria attempt to find the model that best explains the data with a minimum of free parameters. The AIC penalizes free parameters less strongly than does the Schwarz criterion. AIC = 2k + n [ln (SSError / n)] BIC = n ln (SSError / n) + k ln(n)
Two-Factor ANOVA Introduction A method for simultaneously analyzing two factors affecting a response. – Group effect: treatment group or dose level – Blocking factor whose variation can be separated from the error variation to give more precise group comparisons: study center, gender, disease severity, diagnostic group, … One of the most common ANOVA methods used in clinical trial analysis. Similar assumptions as for single-factor anova. Non-parametric alternative : Friedman test
Response score of subject k in column i and row j Overall Mean Effect of treatment factor (a levels or i columns ) Effect of blocking factor (b levels or j rows) Interaction effect Error or Effect of not measured variables Two-Factor ANOVA The Model
Analysis of Covariance ANCOVA Method for comparing response means among two or more groups adjusted for a quantitative concomitant variable, or “covariate”, thought to influence the response. The response variable is explained by independent quantitative variable(s) and qualitative variable(s). Combination of ANOVA and regression. Increases the precision of comparison of the group means by decreasing the error variance. Widely used in clinical trials
Analysis of Covariance The model The covariance model for a single-factor with fixed levels adds another term to the ANOVA model, reflecting the relationship between the response variable and the concomitant variable. The concomitant variable is centered around the mean so that the constant µ represents the overall mean in the model.
Repeated-Measures Basic concepts ‘Repeated-measures’ are measurements taken from the same subject (patient) at repeated time intervals. Many clinical studies require: – multiple visits during the trial – response measurements made at each visit A repeated measures study may involve several treatments or only a single treatment. ‘Repeated-measures’ are used to characterize a response profile over time. Main research question: – Is the mean response profile for one treatment group the same as for another treatment group or a placebo group ? Comparison of response profiles can be tested with a single F-test.
Repeated-Measures Comparing profiles Source: Common Statistical Methods for Clinical Research, 1997, Glenn A. Walker
Repeated Measures ANOVA Random Effects – Mixed Model Prediction Formula What are your conclusions about the between subjects species effect and the within subjects season effect ?
Multiple Regression SPSS Variable Selection Methods Enter. A procedure for variable selection in which all variables in a block are entered in a single step. Forward Selection (Likelihood Ratio). Stepwise selection method with entry testing based on the significance of the score statistic, and removal testing based on the probability of a likelihood-ratio statistic based on the maximum partial likelihood estimates. Backward Elimination (Likelihood Ratio). Backward stepwise selection. Removal testing is based on the probability of the likelihood-ratio statistic based on the maximum partial likelihood estimates.
Cox PH analysis Problem –Can’t use ordinary linear regression because how do we account for the censored data? –Can’t use logistic regression without ignoring the time component with a continuous outcome variable we use linear regression with a dichotomous (binary) outcome variable we use logistic regression where the time to an event is the outcome of interest, Cox regression is the most popular regression technique
Question: When there are many confounding covariates needed to adjust for: –Matching: based on many covariates is not practical –Stratification: is difficult, as the number of covariates increases, the number of strata grows exponentially: 1 covariate: 2 strata 5 covariates: 32 (2 5 ) strata –Regression adjustment: may not be possible: potential problem: over-fitting
Propensity score Replace the collection of confounding covariates with one scalar function of these covariates Age Gender Ejection fraction Risk factors Lesion characteristics … 1 composite covariate: Propensity Score Balancing score
Compare treatments with propensity score Three common methods of using the propensity score to adjust results: – Matching – Stratification – Regression adjustment
Goal of a clinical trial is appraisal of… Superiority: difference in biologic effect or clinical effect Equivalence: lack of meaningful/clinically relevant difference in biologic effect or clinical effect Non-inferiority: lack of meaningful/clinically relevant increase in adverse clinical event
Superiority RCT Possibly greatest medical invention ever Randomization of adequate number of subjects ensures prognostically similar groups at study beginning If thorough blinding is enforced, even later on groups maintain similar prognosis (except for effect of experiment) Sloppiness/cross-over makes arm more similar - > traditional treatment is not discarded Per-protocol analysis almost always misleading
Equivalence/non-inferiority RCT Completely different paradigm Goal is to conclude new treatment is not “meaningfully worse” than comparator Requires a subjective margin Sloppiness/cross-over makes arm more similar -> traditional treatment is more likely to be discarded Per-protocol analysis possibly useful to analyze safety, but bulk of analysis still based on intention-to-treat principle
Superiority, equivalence or non- inferiority? Vassiliades et al, JACC 2005
Possible outcomes in a non-inferiority trial (observed difference & 95% CI) New Treatment Better New Treatment Worse
Typical non-inferiority design Hiro et al, JACC 2009
Cumulative meta-analysis Antman et al, JAMA 1992
Meta-analysis of intervention studies De Luca et al, EHJ 2009
Indirect and network meta-analyses Indirect Direct plus indirect (i.e. network) Jansen et al, ISPOR 2008
Resampling Resampling refers to the use of the observed data or of a data generating mechanism (such as a die or computer-based simulation) to produce new hypothetical samples, the results of which can then be analyzed. The term computer-intensive methods also is frequently used to refer to techniques such as these…
Bootstrap The bootstrap is a modern, computer-intensive, general purpose approach to statistical inference, falling within a broader class of resampling methods. Bootstrapping is the practice of estimating properties of an estimator (such as its variance) by measuring those properties when sampling from an approximating distribution. One standard choice for an approximating distribution is the empirical distribution of the observed data.
Jacknife Jacknifing is a resampling method based on the creation of several subsamples by excluding a single case at the time. Thus, the are only N jacknife samples for any given original sample with N cases. After the systematic recomputation of the statistic estimate of choice is completed, an point estimate and an estimate for the variance of the statistic can be calculated.
The main feature of Bayesian statistics is that it takes into account prior knowledge of the hypothesis
Bayes theorem P (D | H) * P (H) P (H | D) P (D | H) * P (H) _____________ _____________ P (D) P (D) P (H | D) = Likelihood of hypothesis (or conditional probability of B) Prior (or marginal) probability of hypothesis Posterior (or conditional) probability of hypothesis H Probability of the data (prior or marginal probability of B: normalizing constant) Thus it relates the conditional and marginal probabilities of two random events and it is often used to compute posterior probabilities given observations.
“Classical” statistical inference vs Bayesians inference
A Bayesian is who, vaguely expecting a horse, and catching a glimpse of a donkey, strongly believes he has seen a mule Before the next module, a question for you: who is a Bayesian?
A Bayesian is who, vaguely expecting a horse, and catching a glimpse of a donkey, strongly believes he has seen a mule Before the next module, a question for you: who is a Bayesian?
A Bayesian is who, vaguely expecting a horse, and catching a glimpse of a donkey, strongly believes he has seen a mule Before the next module, a question for you: who is a Bayesian?
JMP Statistical Discovery Software JMP is a software package that was first developed by John Sall, co-founder of SAS, to perform simple and complex statistical analyses. It dynamically links statistics with graphics to interactively explore, understand, and visualize data. This allows you to click on any point in a graph, and see the corresponding data point highlighted in the data table, and other graphs. JMP provides a comprehensive set of statistical tools as well as design of experiments and statistical quality control in a single package. JMP allows for custom programming and script development via JSL, originally know as "John's Scripting Language“. An add-on JMP Genomics comes with over 100 analytic procedures to facilitate the treatment of data involving genetics, microarrays or proteomics. Pros: very intuitive, lean package for design and analysis in research Cons: less complete and less flexible than the complete SAS system Price: €€€€.
R R is a programming language and software environment for statistical computing and graphics, and it is an implementation of the S programming language with lexical scoping semantics. R is widely used for statistical software development and data analysis. Its source code is freely available under the GNU General Public License, and pre-compiled binary versions are provided for various operating systems. R uses a command line interface, though several graphical user interfaces are available. Pro: flexibility and programming capabilities (eg for bootstrap), sophisticated graphical capabilities. Cons: complex and user-unfriendly interface. Price: free.
S and S-Plus S-PLUS is a commercial package sold by TIBCO Software Inc. with a focus on exploratory data analysis, graphics and statistical modeling It is an implementation of the S programming language. It features object-oriented programming capabilities and advanced analytical algorithms (eg for robust regression, repeated measurements, …) Pros: flexibility and programming capabilities (eg for bootstrap), user-friendly graphical user interface Cons: complex matrix programming environment Price: €€€€-€€.
SAS SAS (originally Statistical Analysis System, 1968) is an integrated suite of platform independent software modules provided by SAS Institute (1976, Jim Goodnight and Co). The functionality of the system is very complete and built around four major tasks: data access, data management, data analysis and data presentation. Applications of the SAS system include: statistical analysis, data mining, forecasting; report writing and graphics; operations research and quality improvement; applications development; data warehousing (extract, transform, load). Pros: very complete tool for data analysis, flexibility and programming capabilities (eg for Bayesian, bootstrap, conditional, or meta-analyses), large volumes of data Cons: complex programming environment, labyrinth of modules and interfaces, very expensive Price: €€€€-€€€€
Statistica STATISTICA is a powerful statistics and analytics software package developed by StatSoft, Inc. Provides a wide selection of data analysis, data management, data mining, and data visualization procedures. Features of the software include basic and multivariate statistical analysis, quality control modules and a collection of data mining techniques. Pros: extensive range of methods, user-friendly graphical interface, has been called “the king of graphics” Cons: limited flexibility and programming capabilities, labyrinth Price: €€€€.
SPSS SPSS (originally, Statistical Package for the Social Sciences) is a computer program used for statistical analysis released in its first version in 1968 and now distributed by IBM. SPSS is among the most widely used programs for statistical analysis in social science. It is used by market researchers, health researchers, survey companies, government, education researchers, marketing organizations and others. Pros: extensive range of tests and procedures, user-friendly graphical interface. Cons: limited flexibility and programming capabilities. Price: €€€€.
Stata Stata (name formed by blending "statistics" and "data“) is a general-purpose statistical software package created in 1985 by StataCorp. Stata's full range of capabilities includes: data management, statistical analysis, graphics generation, simulations, custom programming. Most meta-analyses tools were first developed for Stata, and thus this package offers one of the most extensive library of statistical tools for systematic reviewers Pros: flexibility and programming capabilities (eg for bootstrap, or meta-analyses), sophisticated graphical capabilities Cons: relatively complex interface Price: €€-€€€
WinBUGS and OpenBUGS WinBUGS (Windows-based Bayesian inference Using Gibbs Sampling) is a statistical software for the Bayesian analysis of complex statistical models using Markov chain Monte Carlo (MCMC) methods, developed by the MRC Biostatistics Unit, at the University of Cambridge, UK. It is based on the BUGS (Bayesian inference Using Gibbs Sampling) project started in 1989. OpenBUGS is the open source variant of WinBUGS. Pros: flexibility and programming capabilities Cons: complex interface Price: free
Advanced statistical methods are best seen as a set of modular tools which can be applied and tailored to the specific task of interest. The concept of generalized linear model highlights how most statistical methods can be considered part of a broader family of methods, depending on the specific framework or link function.
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