Linear models and effect magnitudes for research, clinical and practical applications.
Subject: Linear models (Statistics) (Usage)
Linear models (Statistics) (Analysis)
Linear regression models (Usage)
Linear regression models (Analysis)
Statistics (Models)
Statistics (Equipment and supplies)
Author: Hopkins, Will G.
Pub Date: 01/01/2010
Publication: Name: Sportscience Publisher: Internet Society for Sport Science Audience: Academic Format: Magazine/Journal Subject: Health Copyright: COPYRIGHT 2010 Internet Society for Sport Science ISSN: 1174-9210
Issue: Date: Annual, 2010 Source Volume: 14
Topic: Event Code: 440 Facilities & equipment
Geographic: Geographic Scope: New Zealand Geographic Code: 8NEWZ New Zealand
Accession Number: 297427043
Full Text: * Importance of Effect Magnitudes

* Getting Effects from Models

* Linear models; adjusting for covariates; interactions; polynomials

* Effects for a continuous dependent

* Difference between means; "slope"; correlation

* General linear models: t tests; multiple linear regression; ANOVA...

* Uniformity of error; log transformation; within-subject and mixed models

* Effects for a nominal or count dependent

* Risk difference; risk, odds, hazard and count ratios

* Generalized linear models: Poisson, logistic, log-hazard

* Proportional-hazards regression

Getting Effects from Models

* An effect arises from a dependent variable and one or more predictor (independent) variables.

* The relationship between the values of the variables is expressed as an equation or model.

* Example of one predictor: Strength = a + b*Age

* This has the same form as the equation of a line, Y = a + b*X, hence the term linear model.

* The model is used as if it means: Strength [left arrow] a + b*Age.

* If Age is in years, the model implies that older subjects are stronger.

* The magnitude comes from the "b" coefficient or parameter.

* Real data won't fit this model exactly, so what's the point?

* Well, it might fit quite well for children or old folks, and if so...

* We can predict the average strength for a given age.

* And we can assess how far off the trend a given individual falls.

* With kids, inclusion of Size would reduce the effect of Age. To that extent, Size is a mechanism or mediator of Age.

* But sometimes a covariate is a confounder rather than a mediator.

* Example: Physical Activity (predictor) has a strong relationship with Health (dependent) in a sample of old folk. Age is a confounder of the relationship, because Age causes bad health and inactivity.

* Again, including potential confounders as covariates produces the pure effect of a predictor.

* Think carefully when interpreting the effect of including a covariate: is the covariate a mechanism or a confounder?

* If you are concerned that the effect of Age might differ for subjects of different Size, you can add an interaction.

* Example of an interaction:

Strength = a + b*Age + c*Size+ d*Age*Size

* This model implies that the effect of Age on Strength changes with Size in some simple proportional manner (and vice versa).

* It's still known as a linear model.

Background: The Rise of Magnitude of Effects

* Research is all about the effect of something on something else.

* The somethings are variables, such as measures of physical activity, health, training, performance.

* An effect is a relationship between the values of the variables, for example between physical activity and health.

* We think of an effect as causal: more active [right arrow] more healthy.

* But it may be only an association: more active [left and right arrow] more healthy.

* Effects provide us with evidence for changing our lives.

* The magnitude of an effect is important.

* In clinical or practical settings: could the effect be harmful or beneficial? Is the benefit likely to be small, moderate, large...?

* In research settings:

* Effect magnitude determines sample size.

* Meta-analysis is all about averaging magnitudes of study-effects.

* So various research organizations now emphasize magnitude

* Example of two predictors: Strength = a + b*Age + c*Size

* Additional predictors are sometimes known as covariates.

* This model implies that Age and Size have effects on strength.

* It's still called a linear model (but it's a plane in 3-D).

* Linear models have an incredible property: they allow us to work out the "pure" effect of each predictor.

* By pure here I mean the effect of Age on Strength for subjects of any given Size.

* That is, what is the effect of Age if Size is held constant?

* That is, yeah, kids get stronger as they get older, but is it just because they're bigger, or does something else happen with age?

* The something else is given by the "b": if you hold Size constant and change Age by one year, Strength increases by exactly "b".

* We also refer to the effect of Age on Strength adjusted for Size, controlled for Size, or (recently) conditioned on Size.

* Likewise, "c" is the effect of one unit increase in Size for subjects of any given Age.

* You still use this model to adjust the effect of Age for the effect of Size, but the adjusted effect changes with different values of Size.

* Another example of an interaction:

Strength = a + b*Age + c*Age*Age = a + b*Age + c* [Age.sub.2]

* By interacting Age with itself, you get a non-linear effect of Age, here a quadratic.

* If c turns out to be negative, this model implies strength rises to a maximum, then comes down again for older subjects.

* To model something falling to a minimum, c would be positive.

* To model more complex curvature, add d*[Age.sup.3], e* [Age.sup.4].

* These are cubics, quartics..., but it's rare to go above a quadratic.

* These models are also known as polynomials.

* They are all called linear models, even though they model curves.

* Use the coefficients to get differences between chosen values of the predictor, and values of predictor and dependent at max or min.

* Complex curvature needs non-linear modeling (see later) or linear modeling with the predictor converted to a nominal variable.

* Group, factor, classification or nominal variables as predictors:

* We have been treating Age as a number of years, but we could instead use AgeGroup, with several levels; e.g., child, adult, elderly.

* Stats packages turn each level into a dummy variable with values of 0 and 1, then treat each as a numeric variable. Example:

* Strength = a + b*AgeGroup is treated as Strength = a + [b.sub.1]*Child + [b.sub.2]*Adult + [b.sub.3]*Elderly, where Child=1 for children and 0 otherwise, Adult=1 for adults and 0 otherwise, and Elderly=1 for old folk and 0 otherwise.

* The model estimates the mean value of the dependent for each level of the predictor: mean strength of children = a + [b.sub.1].

* And the difference in strength of adults and children is [b.sub.2] -[b.sub.1].

* You don't usually have to know about coding of dummies, but you do when using SPSS for some mixed models and controlled trials.

* Dummy variables can also be very useful for advanced modeling.

* For simple analyses of differences between group means with t-tests, you don't have to think about models at all!

* Or you can model change scores between pairs of trials. Example:

* Strength = a + b*Group*Trial, where b has four values, is equivalent to StrengthChange = a + b*Group, where b has just two values (expt and cont) and StrengthChange is the post-pre change scores.

* You can include subject characteristics as covariates to estimate the way they modify the effect of the treatment. Such modifiers or moderators account for individual responses to the treatment.

* A popular modifier is the baseline (pre) score of the dependent: StrengthChange = a + b*Group + c*Group*StrengthPre.

* Here the two values of c estimate the modifying effect of baseline strength on the change in strength in the two groups.

* And [c.sub.2] - [c.sub.1] is the net modifying effect of baseline on the change.

* Bonus: a baseline covariate improves precision of estimation when the dependent variable is noisy.

* Modeling of change scores with a covariate is built into the controlled-trial spreadsheets at Sportscience.

Specific Linear Models, Effects and Threshold Magnitudes

* These depend on the four kinds (or types) of variable.

* Continuous: mass, distance, time, current; measures derived there from, such as force, concentration, voltage.

* Counts: such as number of injuries in a season.

* Ordinal: values are levels with a sense of rank order, such as a 4-pt Likert scale for injury severity (none, mild, moderate, severe).

* Nominal: values are levels representing names, such as injured (no, yes), and type of sport (baseball, football, hockey).

* As predictors, the first three can be simplified to numeric.

* If a polynomial is inappropriate, parse into 3-5 levels of a nominal.

* Example: Age becomes AgeGroup (5-14, 15-29, 30-59, 60-79, >79).

* Values can also be parsed into equal quantiles (e.g., quintiles).

* If an ordinal predictor such as a Likert scale has only 2-4 levels, or if the values are stacked at one end of the scale, analyze the values as levels of a nominal variable.

* Linear models for controlled trials

* For a study of strength training without a control group:

Strength = a + b*Trial, where Trial has values pre, post or whatever.

* b*Trial is really [b.sub.1]*Pre + [b.sub.2]*Post, with Pre=1 or 0 and Post=1 or 0.

* The effect of training on mean strength is given by [b.sub.2] -[b.sub.1].

* For a study with a control group:

Strength = a + b*Group*Trial, where Group has values expt, cont.

* b*Group*Trial is really [b.sub.1]*ContPre + [b.sub.2]*ContPost + [b.sub.3]*ExptPre + [b.sub.4]*ExptPost.

* The changes in the groups are given by [b.sub.2] -[b.sub.1] and [b.sub.4] -[b.sub.3].

* The net effect of training is given by (b4-b3) - (b2 -b1).

* Stats packages also allow you to specify this model: Strength = a + b*Group + c*Trial + d*Group*Trial.

* Group and Trial alone are known as main effects.

* This model is really the same as the interaction-only model.

* It does allow easy estimation of overall mean differences between groups and mean changes pre to post, but these are useless.

* You can include the change score of another variable as a covariate to estimate its role as a mediator (i.e., mechanism) of the treatment. Example: StrengthChange = a + b*Group + d*MediatorChange.

* d represents how well the mediator explains the change in strength.

* [b.sub.2] -[b.sub.1] is the effect of the treatment when MediatorChange=0; that is, the effect of the treatment not mediated by the mediator.

* Linear vs non-linear models

* Any dependent equal to a sum of predictors and/or their products is a linear model.

* Anything else is non-linear, e.g., an exponential effect of Age, to model strength reaching a plateau rather than a maximum.

* Almost all statistical analyses are based on linear models.

* And they can be used to adjust for other effects, including estimation of individual responses and mechanisms.

* Non-linear procedures are available but are more difficult to use.

* As dependents, each type of variable needs a different approach. Summary of main effects and models (with examples):

* You consider the difference or change in the mean for pairwise or other comparisons of levels of the predictor.

* Interpret magnitudes using your clinical or practical experience. "

* Otherwise use the standardized difference or change.

* Also known as Cohen's effect size or Cohen's d statistic.

* You express the difference or change in the mean as a fraction of the between-subject standard deviation ([DELTA]mean/SD).

* Use the SD of the reference group or condition, not an average SD.

* Correction factor for bias: 1-3/(4v-1), where v=deg. free. for the SD.

* The smallest important effect is [+ or -]0.2. Complete (+ive) scale:

trivial 0.2 small 0.6 moderate 1.2 large 2.0 very large 4.0 ext. large



* Beware: smallest effect on athletic performance depends on method of measurement, because...

* A percent change in an athlete's ability to output power results in different percent changes in performance in different tests.

* These differences are due to the power-duration relationship for performance and the power-speed relationship for different modes of exercise.

* Example: a 1 % change in endurance power output produces the following changes ...

* 1% in running time-trial speed or time;

* 0.4% in road-cycling time-trial time;

* 0.3% in rowing-ergometer time-trial time;

* 15% in time to exhaustion in a constant-power test.

* A hard-to-interpret change in any test following a fatiguing pre-load.

* Example: the effect of a treatment on strength



* Interpretation of standardized difference or change in means:

* Caution: standardizing works only when the SD comes from a sample representative of a well-defined population.

Measures of Athletic Performance

* For fitness tests of team-sport athletes, use standardization.

* For top solo athletes, an enhancement that results in one extra medal per 10 competitions is the smallest important effect.

* Simulations show this enhancement is achieved with 0.3 of an athlete's typical variability from competition to competition.

* Example: if the variability is a coefficient of variation of 1%, the smallest important enhancement is 0.3%.

* Note that in many publications I have mistakenly referred to 0.5 of the variability as the smallest effect.

* Moderate, large, very large and extremely large effects result in an extra 3, 5, 7 and 9 medals in every 10 competitions.

* The corresponding enhancements as factors of the variability are: trivial 0.3 small 0.9 moderate 1.6 large 2.5 very large 4.0 ext. large

* A slope is more practical than a correlation.

* But unit of predictor is arbitrary, so it's hard to define smallest effect for a slope.

* Example: -2% per year may seem trivial, yet -20% per decade may seem large.

* For consistency with interpretation of correlation, better to express slope as difference per two SDs of predictor.

* It gives the difference between a typically low and high subject.

* See the page on effect magnitudes at for more.

* Easier to interpret the correlation, using Cohen's scale.

* Smallest important correlation is [+ or -] 0.1. Complete scale:

trivial 0.1 small 0.3 moderate 0.5 large 0.7 very large 0.9 ext. large


* The effect of a nominal predictor can also be expressed as a correlation = [square root of (fraction of "variance explained")].

* A 2-level predictor scored as 0 and 1 gives the same correlation.

* With equal number of subjects in each group, the scales for correlation and standardized difference match up.

* For >2 levels, the correlation can't be applied to individuals. Avoid.

* Correlations when controlling for something...

* Interpreting slopes and differences in means is no great problem when you have other predictors in the model.

* Be careful about which SD you use to standardize.

* But correlations are a challenge.

* The correlation is either partial or semi-partial (SPSS: "part").

* Partial = effect of the predictor within a virtual subgroup of subjects who all have the same values of the other predictors.

* Semi-partial = unique effect of the predictor with all subjects.

* Partial is probably more appropriate for the individual.

* Confidence limits may be a problem in some stats packages.

The Error Term in Linear Models with a Continuous Dependent

* Strength = a + b*Age isn't quite right for real data, because no subject's data fit this equation exactly.

* What's missing is a different error for each subject: Strength = a + b*Age + error

* This error is given an overall mean of zero, and it varies randomly (positive and negative) from subject to subject.

* It's called the residual error, and the values are the residuals.

* residual = (observed value) minus (predicted value)

* In many analyses the error is assumed to have values that come from a normal (bell-shaped) distribution.

* This assumption can be violated a lot. Testing for normality is not an issue, thanks to the Central Limit Theorem.

* With a count as the dependent, the error has a Poisson (or the related negative binomial) distribution, which is an issue

* Address with general/zed linear modeling-see later.

* Address the problem by analyzing the log-transformed dependent.

* 5% effect means Post = Pre*1.05.

* Therefore log(Post) = log(Pre) + log(1.05).

* That is, the effect is the same for everyone: log(1.05).

* And we now have a linear (additive) model, not a non-linear model, so we can use all our usual linear modeling procedures.

* A 5% error means typically x1.05 and / 1.05, or x//-1.05.

* And a 100% error means typically x//2.0 (i.e., values vary typically by a factor of 2), and so on.

* When you finish analyzing the log-transformed dependent, you back-transform to a percent or factor effect.

* Show percents for anything up to 30%. Show factors otherwise, e.g., when the dependent is a hormone concentration.

* Use the log-transformed values when standardizing.

* Log transformation is often appropriate for a numeric pred/ctor.

* The effect of the predictor is then expressed per percent, per 10%, per 2-fold increase, and so on.

The Names of Linear Models with a Continuous Dependent

* You need to know the jargon so you can use the right procedure in a spreadsheet or stats package.

* Unpaired t test: for 2 levels of a single nominal predictor.

* Use the unequal-variances version, never the equal-variances.

* Paired t test: as above, but the 2 levels are for the same subjects.

* Simple linear regression: a single numeric predictor.

* Multiple linear regression: 2 or more numeric predictors.

* Analysis of variance (ANOVA): one or more nominal predictors.

* Analysis of covariance (ANCOVA): one or more nominal and one or more numeric predictors.

* Repeated-measures analysis of (co)variance: AN(C)OVA in which each subject has two or more measurements.

* General linear model (GLM): any combination of predictors.

* In SPSS, nominal predictors are factors, numerics are covariates.

* Mixed linear model: any combination of predictors and errors.

* You characterize the error with a standard deviation.

* It's also known as the standard error of the estimate or the root mean square error.

* In general linear models, the error is assumed to be uniform.

* That is, there is only one SD for the residuals, or the error for every datum is drawn from a single "hat".

* Non-uniform error is known as heteroscedasticity.

* If you don't do something about it, you get wrong answers.

* Without special treatment, many datasets show bigger errors for bigger values of the dependent.

* This problem is obvious in some tables of means and SDs, in scatter plots, or in plots of residual vs predicted values (see later).

* Such plots of individual values are also good for spotting outliers.

* It arises from the fact that effects and errors in the data are percents or factors, not absolute values.

* Example: an error or effect of 5% is 5 s in 100 s but 10 s in 200 s.


* Rank transformation is another way to deal with non-uniformity.

* You sort all the values of the dependent variable, then rank them (i.e., number them 1, 2, 3,...).

* You then use this rank in all further analyses.

* The resulting analyses are sometimes called non-parametric.

* But it's still linear modeling, so it's really parametric.

* They have names like Wilcoxon and Kruskal-Wallis.

* Some are truly non-parametric: the sign test; neural-net modeling.

* Some researchers think you have to use this approach when "the data are not normally distributed".

* In fact, the rank-transformed dependent is anything but normally distributed: it has a uniform (flat) distribution!!!

* So it's really an approach to try to get uniformity of effects and error.

* Problems: it doesn't necessarily give uniformity; you lose a lot of information; it's hard to convert the rank effects back to raw values.

* So use ranks as a last resort.

* Mixed modeling is the cutting-edge approach to the error term.

* Mixed = fixed effects + random effects.

* Fixed effects are the usual terms in the model; they estimate means.

* Fixed, because they have the same value for everyone in a group or subgroup; they are not sampled randomly.

* Random effects are error terms and anything else randomly chosen from some population; each is summarized with an SD.

* The general linear model allows only one error. Mixed models allow:

* specification of different errors between and within subjects;

* within-subject covariates (GLM allows only subject characteristics or other covariates that do not change between trials);

* specification of individual responses to treatments and individual differences in subjects' trends;

* interdependence of errors and other random effects, which arises when you model different lines or curves for each subject.

* With repeated measurement in controlled trials, simplify analyses by analyzing change scores, even when using mixed modeling.

* Relative risk or risk ratio = a/b.

* Example: 83/50 = 1.66 or "66% increase in risk".

* Widely used but inappropriate for common time-dependent events.

* Hazards and hazard ratios are better: see later.

* For rare events, risk ratio is OK, because same as hazard ratio.

* Can't estimate directly with linear models.

* Risk difference or odds ratio are better for common classifications.

* Magnitude scale: use risk difference, odds ratio or hazard ratio. For the experts:

* Number needed to treat (NNT) = 100/(a- b).

* = number you would have to treat or sample for one subject to have an outcome attributable to the effect.

* Promoted in some clinical journals, but not widely used?

* Can't estimate directly with linear models.

* Magnitude scale (if you ever use it) is given by 1/(risk difference).


* Non-uniformity also arises with different groups and time points.

* Example: a simple comparison of means of males and females, with different SD for males and females (even after log transformation).

* Hence the unequal-variances t statistic or test.

* To include covariates here, you can't use the general linear model: you have to keep the groups separate, as in my spreadsheets.

* Example: a controlled trial, with different errors at different time points arising from individual responses and changes with time.

* MANOVA and repeated-measures ANOVA can give wrong answers.

* Address by reducing or combining repeated measurements into a single change score for each subject: within-subject modeling.

* Then allow for different SD of change scores by analyzing the groups separately, as above.

* Bonus: you can calculate individual responses as an SD.

* See Repeated Measures and Random Effects at and/or the article on the controlled-trial spreadsheets for more.

* Or specify several errors and much more with a mixed model...

* For time-dependent effects, subjects start "N" 10 but different proportions end up "Y".

* Risk or proportion difference = a - b.

* Example: a-b = 83%-50% = 33%, so at the time point shown, an extra 33 of every 100 males are injured because they are male.

* Good for common events, but time-dependent.

* Can't model risks and estimate differences directly in linear models.

* Smallest effect: 10% at time when the risk difference is maximum.

* At that time, 1 male in every 10 is injured due to being male.

* Complete scale (for common events, where everyone gets affected): trivial 10% small 30% moderate 50% large 70% very large 90% ext. large

* This scale applies also to time-independent common classifications.


* Odds ratio = (a/c)/(b/d).

* Hard to interpret, but must use to express effects and confidence limits for time-independent classifications, including some case-control designs.

* Use hazard ratio for time-dependent risks.

* Magnitudes for common time-independent Classifications ...

* Either convert to difference in risk between the reference (comparison or control) group and other group. Example shown: if 50% of reference group is affected, and odds ratio is 4.9, then by simple algebra, 83% of other group is affected. Therefore risk difference = 33% (i.e., moderate).

* Or use this scale for odds ratios, which correspond to risk differences of 10, 30, 50, 70 and 90% "centered" on 50%:_

trivial 1.5 small 3.4 moderate 9.0 large 32 very large 360 ext. large

* Magnitudes for rare classifications: see later.


* Hazard ratio or incidence rate ratio = e/f.

* Hazard = instantaneous risk rate = proportion per infinitesimal of time.

e = 100 %/5wk = 20 %/wk = 2.9 %/d f = 40 %/5wk = 8 %/wk = 1.1 %/d - f e/f= 100/40 = 20/8 = 2.9/1.1 = 2.5

* Hazard ratio is the best statistical measure Time (months) for time-dependent events.

* It's the risk ratio right now: male risk is 2.5x the female risk.

* Effects and confidence limits can be derived with linear models.

* Obviously not dependent on time if hazard rates are constant.

* And even if both rates change, often OK to assume their ratio is constant, which is the basis of proportional hazards regression.

* Magnitude scale depends on whether event is common or rare.

* For common events and constant hazards, maximum risk differences translate into hazard ratios of 1.3, 2.3, 4.4, 10, and 50.

* Hazard ratios for rare events: see later.

Magnitude Thresholds for Rare Events and Classifications

* The focus is the affected few and/or those who deal with them.

* Hazard ratio = risk ratio = odds ratio for low risks (or short times).

* A ratio of 1.1 would produce a 10% increase or decrease in the workload of anyone dealing with the event.

* Anything less might go unnoticed, so 1.1 is the smallest effect.

* Or in a group of affected individuals, 1 in 10 able to blame the effect represents a defensible smallest effect. Similarly 3, 5, 7 and 9 individuals in 10 represent the other magnitude thresholds.

* Corresponding hazard ratios are 10/9, 10/7, 10/5, 10/3, and 10/1. Hence...

* Hazard-ratio thresholds for rare events:

trivial 1.1 small 1.4 moderate 2.0 large 3.3 very large 10 ext. large

* By a similar argument, this scale applies to count ratios.

* Oddly, these thresholds are smaller than those for common events.

* We tolerate relatively higher risk for a rare event, but if we end up as an event, we wish the risk had been lower!

* Effect of a nominal predictor is expressed as a ratio (factor) or percent difference.

* Example: in their sporting careers, women get 2.3 times more tendon injuries than men.

* If the ratio is 1.5 or less, it can be expressed as a percent: men get 26% (1.26 times) more muscle sprains than women.

* Effects of a numeric predictor are expressed as factors or percents per unit or per 2 SD of the predictor.

* Example: 13% more tackles per 2 SD of repeated-sprint speed.

* Magnitude scale for count ratios is same as for rare events: trivial 1.1 small 1.4 moderate 2.0 large 3.3 very large 10 ext. large

* Ratio of mean time to event = [t.sub.2]/[t.sub.1]. 100

* Easier for an individual to interpret.

* If the hazards are constant, it's also the inverse of the hazard ratio.

* Example: if hazard ratio is 2.5, there is 2.5x the risk of injury. But 1/2.5 = 0.4, so injury occurs in less than half the time, on average.

* Difference in mean time to event = [t.sub.2]-[t.sub.1]

* Also easy to interpret, but can't model directly.

* Standardization of the log of individual values of time to event leads to another scale for hazard ratios or mean-time ratios of common events: 1.3, 2.2, 4.5, 13, 100.

* This scale is similar to that given by consideration of maximum risk difference for common events. Averaging the two and simplifying ...

* Hazard-ratio thresholds for common events:

trivial 1.3 small 2.3 moderate 4.5 large 10 very large 100 ext. large


* Derive and interpret the "slope" (a correlation isn't defined here).

* As with a nominal predictor, you have to express effects as odds or hazard ratios (for time-independent or -dependent events) to get confidence limits.

* Example shows how chances would change with fitness, and the meaning of the odds ratio per unit of fitness: (b/d)/(a/c).

* Odds ratio here is (75/25)/(25/75) = 9.0 per unit of fitness.

* Best to express as odds or hazard ratio per 2 SD of predictor.

* Magnitude scales are then the same as for nominal predictors.


Details of Linear Models for Events, Classifications, Counts

* Counts, and binary variables representing levels of a nominal, give wrong answers as dependents in the general linear model.

* It can predict negative or non-integral values, which are impossible.

* Non-uniformity would also be an issue.

* Generalized linear modeling has been devised for such variables.

* The generalized linear model predicts a dependent that can range continuously from -[infinity] to +[infinity], just as in the general linear model.

* You specify the dependent by specifying the distribution of the dependent and a link function.

* For a continuous dependent, specifying the normal distribution and the identity link produces the general linear model.

* Don't use this approach with continuous dependents, because the standard procedures for general linear modeling are easier.

* Easiest to understand the approach with counts first...

* For counts (e.g., each athlete's number of injuries), the dependent is the log of the mean count.

* The mean count ranges continuously from 0 to +[infinity].

* The log of the mean count ranges from -[infinity]> to +[infinty".

* So the link function is the log.

* Specify the distribution for counts, Poisson.

* The model is called Poisson regression.

* The log link results in effects expressed as count ratios.

* If the counts accumulate over different periods for different subjects, you can specify the period in the model as an offset or denominator.

* You are then modeling rates, and the effects are rate ratios.

* Specify a negative binomial distribution if you think the events for each subject tend to occur in clusters rather than truly randomly.

* The model thereby reflects the fact that the counts have a bigger variation for a given predicted count than purely Poisson counts.

* For binary variables representing time-dependent events (e.g., un/injured), the dependent is the log of the hazard.

* The hazard is the probability of the event per unit time.

* For events that accumulate with a constant hazard (h), the proportion of subjects affected at time t is given by

p = 1 - [e.sup.-h.t*]; hence h = -log(1 - p).

* The hazard ranges continuously from 0 to +00.

* Log of the hazard ranges from -a> to +".

* The link function is known confusingly as the complementary log- log: log(-log(1-p)).

* I prefer to refer to the log-hazard link.

* Specify the distribution for binary events, binomial.

* The model has no common name. I call it log-hazard regression.

* The log of the hazard results in effects expressed as hazard ratios.

* You can specify a different monitoring time for each subject.

* When hazards aren't constant, use proportional hazards regression.

* When the dependent is a nominal with >2 levels, group into various combinations of 2 levels and use the above models, or...

* For the advanced class ...

* Multinomial logistic regression, for time-independent nominals (e.g., a study of predictors of choice of sport).

* Use the multinomial distribution and the generalized logit link (available in SAS in the new Glimmix procedure).

* SAS does not provide a link in Glimmix or Genmod for multinomial hazard regression of time-dependent nominals.

* Cumulative logistic regression, for time-independent ordinals (e.g., injury severity on a 4-point Likert scale).

* Multinomial distribution; cumulative logit link.

* Use for <5-pt or skewed Likert scales; otherwise use general linear.

* Cumulative hazard regression, for time-dependent ordinals (e.g., uninjured, mild injury, moderate injury, severe injury).

* Multinomial distribution; cumulative complementary log-log link.

* Generalized linear models for repeated or clustered measures are also known as generalized estimating equations.

* For binary variables representing time-independent events (e.g., a classification, such as selected or not), the dependent is the log of the odds of the event occurring.

* Odds=p/(1-p), where p is the probability of the event.

* P ranges from 0 to 1, so odds range continuously from 0 to +[infinity].

* So log of the odds ranges from -[infinity]> to +[infinity]".

* So the link function is the log-odds, also known as the logit.

* Specify the distribution for binary events, binomial.

* The model is called logistic regression, but log-odds regression would be better.

* The log of the odds results in effects expressed as odds ratios.

* A log-odds model may be simplistic or unrealistic, but it's got to be better than modeling p or log p, which definitely does not work.

* Some researchers mistakenly use this model for time-dependent events, such as development of injury. But ...

* If proportions of subjects experiencing the event are low, you can model risk, odds or hazards, because the ratios are the same.

Three slides for the experts Other Models for Events and Classifications

*All have outcomes modeled as ratios (between levels of nominal predictors) or ratios per unit (or per 2 SD) of numeric predictors.

*The magnitude scales for common and rare events and classifications are the same as in previous models.

* Summary (with examples):

* Proportional hazards (Cox) regression is another and more advanced form of linear modeling for time-dependent events.

* Use when hazards can change with time, if you can assume ratios of the hazards of the effects are constant.

* Example: hazard changes as the season progresses, but hazard for males is always 1.5x that for females.

* A constant ratio is not obvious in this kind of figure.

* Time to the event is the dependent, but effects are estimated and interpreted as hazard ratios.

* The model takes account of censoring: when someone leaves the study (or the study stops) before the event has occurred.

* Not covered in this presentation: magnitude thresholds for measures of reliability, validity, and diagnostic accuracy.


Main Points

* An effect is a relationship between a dependent and predictor.

* Effect magnitudes have key roles in research and practice.

* Magnitudes are provided by linear models, which allow for adjustment, interactions, and polynomial curvature.

* Continuous dependents need various general linear models.

* Examples: t tests, multiple linear regression, ANOVA...

* Within-subject and mixed modeling allow for non-uniformity of error arising from different errors with different groups or time points.

* Effects for continuous dependents are mean differences, slopes (expressed as 2 SD of the predictor), and correlations.

* Thresholds for small, moderate, large, very large and extremely large standardized mean differences: 0.2, 0.6, 1.2, 2.0, 4.0.

* Thresholds for correlations: 0.1, 0.3, 0.5, 0.7, 0.9.

* Many dependent variables need log transformation before analysis to express effects and errors as uniform percents or factors.

* Counts and nominal dependents (representing classifications and time-dependent events) need various generalized linear models.

* Examples: Poisson regression for counts, logistic regression for classifications, log-hazard regression for events.

* The dependent variable is the log of the mean count, the log of the odds of classification, or the log of the hazard (instantaneous risk) of the event.

* Effect-magnitude thresholds for counts and nominal dependents:

* Percent risk differences for classifications: 10, 30, 50, 70, 90.

* Corresponding odds ratios for classifications: 1.5, 3.4, 9.0, 32, 360.

* Hazard-ratio thresholds for common events: 1.3, 2.3, 4.5, 10, 100.

* Ratio thresholds for counts and rare events: 1.1, 1.4, 2.0, 3.3,10 (apply equally to count, hazard, risk or odds ratios).

* Use proportional hazards regression when hazards vary with time but hazard ratio is constant.

Will G. Hopkins

AUT University, Auckland, NZ (
Dependent             Predictor         Effect of predictor

continuous Strength   nominal Trial     difference in means

continuous Activity   numeric Age       "slope" (difference
                                        per unit of predictor);

nominal InjuredNY     nominal Sex       differences or ratios
                                        of proportions, odds,
                                        rates, hazards

nominal SelectedNY    numeric Fitness   "slope" (difference or
                                        ratio per unit of

count Injuries        nominal Sex       ratio of counts

count Medals          numeric Cost      "slope" (ratio per unit
                                        of predictor)

Dependent             Statistical model

continuous Strength   (un)paired t test; (multiple
                      linear) regression; ANOVA;
                      ANCOVA; general linear; mixed

continuous Activity

nominal InjuredNY     logistic regression;
                      log-hazard regression;
                      generalized linear;

nominal SelectedNY

count Injuries        Poisson regression;
                      generalized linear;

count Medals

Dependent    Predictor   Effect
continuous   nominal     difference or change in means
Strength     Trial

Cohen     Hopkins

        trivial     <0.2       <0.2
          small    0.2-0.5    0.2-0.6
       moderate    0.5-0.8    0.6-1.2
          large      >0.8     1.2-2.0
     very large       ?       2.0-4.0
extremely large       ?        >0.4

Dependent    Predictor   Effect
continuous   numeric     "slope" (difference per
                         unit of predictor);
Activity     Age

Dependent    Predictor     Effect
nominal      nominal       differences or ratios
                           of proportions, odds, rates,
InjuredNY    Sex           hazards, mean event time

Dependent     Predictor         Effect
count         nominal           ratio of counts
Injuries      Sex
count         numeric           "slope" (ratio per unit of predictor)
Tackles       Fitness

Dependent     Predictor      Effect

nominal       numeric       "slope"

Selected      NY Fitness    (difference or
                            ratio per unit
                            of predictor)

Dependent               Effect of predictor    Statistical model

multiple proportions    odds ratio             multinational logistic
choice of several                              regression; generalized
sports                                         linear

ordinal                 odds or hazard         cumulative logistic
injury severity         ratio                  or hazard regression;
(4-pt Likert)                                  generalized linear

time to event           hazard ratio           proportional hazards
time to injury                                 (Cox) regression
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