error bars; power and sample size - university of iowapower and sample size summary introduction in...

Presenting results with error barsPower and sample size

Summary

Error bars; Power and sample size

Patrick Breheny

March 27

Patrick Breheny Introduction to Biostatistics (171:161) 1/29


Summary

Introduction

In this lecture, we’ll discuss two topics that come up in one-samplestudies (as well as other types of studies):

One is the presentation of results, and specifically, thedecision when making figures and tables to include error barsand whether the error bar should be based on the standarddeviation or the standard error

The other is the issue of planning a study, and determiningbeforehand the power of the study for a given sample size



Summary

A dynamite plot

Figures like the following areincredibly common:

Far Near

IQ

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There are a number ofpossible variations, but thegeneral idea is to have a baror a dot or a line that showsthe mean, then error barsthat show . . . something

These particular error barsare one SD, so theyillustrate something aboutvariability of individuals’ IQsin the two samples



Summary

Box plot

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IQ

Keep in mind, however, thatthe “dynamite” plot is justshowing two numbers, themean and SD

This has the potential tomisrepresent the actualdistribution of data, andmore informative sorts ofplots, such as the box plot,are alternatives worthconsidering



Summary

Strip plotIQ

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We could also just show allthe data

This type of plot is knownas a strip plot



Summary

±1 SD again

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IQ

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This is the same as the firstplot, although with dotsreplacing the bars; this hasthe advantage that the bardoesn’t block the lower errorbar, so we can see the±1 SD region more clearlyRecall that the mean ±1 SDtypically contains about themiddle two-thirds of thedata



Summary

±1 SE

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Far Near

Here, we’re plotting ±1 SEinstead of ±1 SDThe error bars are a lotsmaller here, of course, sinceSE = SD/

√n, and now say

something about thevariability of the samplemean or, if you prefer, theerror with which we havemeasured the sample mean



Summary

Error bars: SD vs. SE

So, the ±SD error bar plot emphasized the variability of dataThis doesn’t really have anything to do with any sort of“error” at all, and furthermore, other kinds of plots are usuallybetter for showing the distribution of your data

On the other hand, the ±SE error bar plot emphasizesuncertainty (or possible error) about the mean

However, this isn’t a particularly meaningful plot either, since±SE is only a 68% CI, and really, it isn’t even that, since aswe learned in the previous lecture, the coverage of the CIdepends on the degrees of freedom with which we’vemeasured the SD



Summary

CI plot

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Far Near

So typically, the mostmeaningful route is to plotthe confidence interval

For example, the plot on theleft suggests that it isn’tunreasonable to think thatthe average IQ for bothgroups is 90



Summary

Variability and uncertainty are both of interest

It should be noted, however, that the confidence interval plot,although very useful and informative, doesn’t really tell usanything about the distribution of the data, which is alsointeresting

In principle, this is the choice one makes in deciding on afigure, whether to emphasize uncertainty about the average orto emphasize variability/diversity, although there are ways toillustrate both in a single figure



Summary

Illustrating both variability and uncertainty

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In this plot, the shaded regiondenotes the 95% confidenceinterval, and we can get a senseof the variability amongindividuals as well as theuncertainty about the populationmeans



Summary

±1 SD for NHANES height

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Hei

ght

Male Female

Just to make sure the point isclear, let’s look at the NHANESheight data; here are the means±SD



Summary

CI plot for NHANES height

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ght

Male Female

And here the error bars denotethe 95% confidence interval;both this plot and the previousone tell us different things

It is obvious that men are,on average, taller thanwomen

At the same time, it is justas obvious that there areplenty of women in thepopulation who are tallerthan many men



Summary

NHANES height data

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Summary

Cystic Fibrosis

One last comment: plottingseparate confidence intervals andlooking for overlap is not thesame as plotting a confidenceinterval for the difference

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The disparity is particularlydrastic for paired studies, but alsooccurs for two-sample studies, aswe will see in a week or two

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Summary

Planning a study

One of the most important questions as far as planning andbudgeting a study is concerned is: how many subjects do Ineed?

The number of subjects tends to play a very large role indetermining the cost of a study, so funding agencies generallywant to know the number of subjects that a study will requirebefore they make a decision about whether or not to pay for it

But of course, the fewer subjects you have, the harder it is todistinguish a real phenomenon from chance



Summary

Power

The probability that you will successfully distinguish the realphenomenon from chance (i.e., reject the null hypothesis) iscaptured in the notion of power

Power is the opposite of the type II error we discussed back atthe beginning of the semester

Power is the probability of rejecting the null hypothesis giventhat it is in reality false; the type II error rate (β) was theprobability of failing to reject the null hypothesis given that itwas false

Thus, by the compliment rule:

Power = 1− β



Summary

Two important questions

With the time remaining in today’s lecture, I want to addresstwo highly related questions:

If I plan a study with a certain number of subjects, what is mypower going to be?If I want to achieve a certain power, how large does my samplesize need to be?



Summary

Two important questions (cont’d)

These are important questions for any kind of data, and eachtype of study has its own formulas and procedures forcalculating power

We won’t get into the details of power calculations in thisclass (which can be quite complicated)

Instead, we will focus on the main concepts, which aregenerally similar for any type of study



Summary

Power: Basement analogy

Consider the following analogy1: you send a child into thebasement to find an object

What’s the probability that she actually finds it?

This depends on three things:

How long does she spend looking?How big is the object?How messy is the basement?

1This analogy comes from Intuitive Biostatistics, which in turn credits JohnHartung for the original idea



Summary

Power: Basement analogy (cont’d)

If the child spends a long time looking for a large object in aclean, organized basement, then she’ll probably find it

If the child spends a short time looking for a small object in amessy basement, then there’s a good chance she won’t find it

All three of these questions have statistical analogs:

How long does she spend looking? = How big was the samplesize?How big is the object? = How large is the effect size?How messy is the basement? = How noisy/variable is thedata?



Summary

Specifying effect size and variability

In general, one does not know the effect size or the variability– especially before we have conducted the study

So, in order to calculate power, we are going to essentiallymake up values for these quantities, and our calculated powerwill depend on the values that we choose

Of course, if we specify values that are far away from reality,our power calculations are not going to be accurate

Sometimes, reasonable values for certain quantities can bechosen on the basis of past studies or observations

Other times, a small initial study called a “pilot study” isconducted in order to provide some data with which toestimate these quantities and help plan for a larger study thatwould take place in the future.



Summary

Example

Suppose we develop some intervention that we think canreduce the LDL cholesterol levels of individuals whoparticipate in it

Suppose we plan to conduct a study in which individuals tryboth the intervention and a control, and we are going to lookat the difference in each individual’s LDL cholesterol levels onand off the intervention

The power of our study (the probability that we will get ap-value under .05) depends on:

The sample size (how many people we enroll in our study)The variability (how much variability there is in a person’s LDLcholesterol levels)The effect size (the amount by which our intervention actuallyreduces cholesterol)



Summary

Power curves

The main concepts behind power and sample size can beillustrated in power curves – graphs of what happens to poweras we change one of sample size/variability/effect size

In the curves that follow, I will start with:

A sample size of 100A variability of 36 mg/dLAn effect size of 5 mg/dL

And we will see what happens to power as we vary each ofthem



Summary

Power curve #1

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Summary

Power curve #2

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Summary

Power curve #3

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Summary

Sample size determination

Thus, we can determine our sample size by looking at thepower curve

For instance, in the previous example, if we want a power of80%, we would need a sample size of about 400

In reality, of course, lots of other things like money, time,resources, availability of subjects, etc., influence the actualsample size of a study

Also, we may be interested in calculating the required samplesize under a few different designs to see which way is theeasiest/cheapest to conduct the study



Summary

Summary

Displaying ±SD in a figure emphasizes variability, whiledisplaying ±SE emphasizes uncertainty, although ask yourself:Wouldn’t I be better off plotting the confidence interval?

The power of a study depends on:

Sample sizeVariabilityEffect size


Presenting results with error barsPower and sample sizeSummary

error bars; power and sample size - university of iowapower and sample size summary introduction in...

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