Area under the curve: Continuous data types

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Understanding Clinical Research: Behind the Statistics

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University of Cape Town

Understanding Clinical Research: Behind the Statistics

907 classificações

If you’ve ever skipped over`the results section of a medical paper because terms like “confidence interval” or “p-value” go over your head, then you’re in the right place. You may be a clinical practitioner reading research articles to keep up-to-date with developments in your field or a medical student wondering how to approach your own research. Greater confidence in understanding statistical analysis and the results can benefit both working professionals and those undertaking research themselves. If you are simply interested in properly understanding the published literature or if you are embarking on conducting your own research, this course is your first step. It offers an easy entry into interpreting common statistical concepts without getting into nitty-gritty mathematical formulae. To be able to interpret and understand these concepts is the best way to start your journey into the world of clinical literature. That’s where this course comes in - so let’s get started! The course is free to enroll and take. You will be offered the option of purchasing a certificate of completion which you become eligible for, if you successfully complete the course requirements. This can be an excellent way of staying motivated! Financial Aid is also available.

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Building an intuitive understanding of statistical analysis

There is hardly any healthcare professional who is unfamiliar with the p-value. It is usually understood to have a watershed value of 0.05. If a research question is evaluated through the collection of data points and statistical analysis reveals a value less that 0.05, we accept this a proof that some significant difference was found, at least statistically.In reality things are a bit more complicated than that. The literature is currently full of questions about the ubiquitous p-vale and why it is not the panacea many of us have used it as. During this week you will develop an intuitive understanding of concept of a p-value. From there, I'll move on to the heart of probability theory, the Central Limit Theorem and data distribution.

P-values: P is for probability1:56

Working out the probability: Rolling dice5:24

Area under the curve: Continuous data types4:59

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Juan H Klopper
Dr
Department of Surgery

Now we're going to get to the area under plot, or under graph for

continuous variables.

Now remember with the discrete, we make the width of those little bars one.

It's discrete variables.

They stand on their own.

Now for continuous data by definition we can continuously divide it smaller,

smaller, smaller, smaller and in theory at least there should be no infinite limit.

Now what can we do?

How can we solve this problem?

Because we still want to deal with area.

Can we draw an ultra thin little bar?

Tiny, tiny little base.

Tiny little widths.

We really can't do that.

Because any one you draw, I can cut in half, as well.

By definition.

So remember although that theoretical practically it's not really so,

remember we talked about the blood pressure that if I had

an apparatus that was, could do infinitely fine measurements,

I could work out to a systolic blood pressure of 120.56789 or if I

were dealing with really large numbers like the white cell count in human blood.

practically speaking we can make those bars smaller and smaller and

eventually you're going to end up with a bar with a width of zero.

Then we can't work out data anymore.

So what do we do?

Fortunately mathematicians to the rescue and

as per usual if you see the graph there it's negative X minus H squared.

A nice little graph there.

A parabolic curve there and we can work out the area under the curve.

We needn't have a special shape like a square or rectangle.

The very smooth curve, we can work out the area under the curve for that.

Between two limits.

Integration is quite easy to do.

Now, the mathematics behind

statistics is going to draw this beautifully symmetrical normal curve.

Bell shaped curve for us.

And it can work out.

The area under the curve for that.

Now, you can ask yourself, how would this really work?

Can I just draw a line up from the X axis, and just read off the probability?

Remember that's not how a P value works.

That little bar has a width of zero.

So we can't just look at one specific value, we do something different.

I want to remind you when we roll the dice,

we said what was the probability of rolling an 11 or 12.

So we added those values, and

we're going to do exactly the same with a continuous variable.

We're not, we're never gonna ask what the probability of finding one value is.

We begin to colour in an area under the, the graph for

some distance from the spot out to one of the two sides.

And we are going to calculate that probability.

We're going to find our value somewhere in there and then say if it's more or

less than say, the point of five, that was our magical value.

Now we construct the graph, at least the mathematics or the program that

you use is going to construct that graph so that the whole area is still one.

It's going to be same as we had with the rolling of the die.

Total probability has got to equal one, so it's got to add up to one.

And this is what's going to happen behind the scenes when you do the maths.

If you choose 0.05 now the graph that you see there is not really to scale

but it is going to decide where to make the cutoff so

that you see the darker area, the darker blue area, where that would represent

five percent or 0.05 of the total area under that curve.

The value that is found by doing research is then going to fall

either within those limits and we can work out the p value.

The probability for that area under the curve.

But we would say it is less than 0.05 having found that result with that

research with the probability of less than 0.05.

So it's going to convert all the differences in means, the means of a group.

It's going to convert that to some value on the x-axis, and

its gonna fall either within that 0.05, or it's going to fall outside of it and

we can actually work out that area under the curve.

Now, you've seen me draw little curves and everything was on the one side,

but most of the time that's not how we're going to do it.

We can do this.

We're going to split the 5%, 0.05 up into 0.025.

2.5% on either side and our values are going to fall on either side of that.

We're still going to calculate an area under the curve.

So that's how it works. It's an area under the curve,

it falls between different limits and

the results that are found either falls within those areas or

it falls outside of the p value of less than 0.05 or a p value of more than 0.05.

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