Well, welcome today. I have Rafael PEREIRA with me, who the university lecturer in statistics at the Department of Primary Care Health Sciences.
Good day to you. Good morning.
So Rafael's expertise in statistics and runs the monitoring group here in Oxford and has been working in primary care health care statistics.
And it's a good friend of mine for some 15 years now.
So we're going to look today at three different papers while we have this discussion.
The first of them is absence of evidence is not evidence of absence by Doug Altman.
The second is a really interesting paper which does date quite a bit.
1763 An Effort towards solving a problem problem in the Doctrine of Challenges by the late Reverend Mr. Bays.
And final paper is actually generalised linear models.
So where we're really going to see how we get on with that. So let me start.
Let's say I'm a up and coming student.
I'm doing a Ph.D. and I've come to see a statistician. What do you think are the best ways to prepare?
What do you think people should really know? And statistics before they say come to you?
They may be an epidemiologist and maybe a clinician, but what the sort of problems you see where you think,
well, you could have actually solved that yourself?
I think I think the main issue that we deal with day to day is that people believe that statistics is mainly about a series
of techniques that you would employ once you have your data that will give you an answer or the answer that you want.
And more and more fortunately people are beginning to realise that it's not about that.
So statistics is really a way of thinking. It's more about trying to understand how the data is generated and in because of the same reason,
what sort of data would be the most appropriate to answer a particular question.
So really we want to be involved or people need to be thinking about statistics or if you want study, design, whatever you want to call it,
from the very early phase of your study and you want to think about what are the reasons why you want to collect information,
what's the best possible information you want you possibly can collect to answer the the the question that you have in mind.
And once you have that, if you if you're thinking adequately early on, then the methods you're going to use could be very simple methods.
Sometimes they have to be slightly more complex, sometimes they have to be slightly more technical.
But generally, if you have your question set up, right, gathering data right,
then the methods that you're going to employ will be relatively simple ones.
So let me come on on that point because I think over the years I've seen this problem a lot.
I'd say it's one of the major problems. People say, I have a bit of data.
Yeah, can I see a statistician? Yeah. And actually the issue is you have some data.
And most of the time, as we try to say, well, what question do you want to ask you this?
That's correct. So the onus is it's almost reciprocal.
So if you start early on thinking, okay, what are the data? What data?
I'm going to be collecting too much the right question, then the methods you're going to be employing are going to be very simple.
When it becomes really complex is when people suddenly around say, I've been collecting data for the last 20 years, this is my large dataset.
I want to do something using, Well, what do you want to do?
What are the questions that you want to address? And sometimes this was very, very good advice that I got from Patil.
Who was it? The statistician. When I first arrived to this apartment, she said, Well, first things you need to ask is, what's your question?
And very often you find that when people come in with data, they don't have a question.
And that's that's crucial because I have a statistician, sometimes I have my own questions,
but they may not necessarily match the questions that the clinician or epidemiologists would be wanting to answer.
So I guess this is just relate this to the sort of old trials and making old trial data available.
This is a real concern of people who say, well, if we make data available, people are just going to do what they call dredging.
Yeah. Which I guess is what is the real concern when you have lots of data available and to to to people out there.
What does that so I mean this term you see often bandied about for data dredging more so data dredging is mainly about and it's it's seen as a
very pejorative term but I'll come back to why nowadays is not as bad or has been rebranded into something that is being called now data mining.
But data dredging is mainly about exploring information and exploring data without a clear idea what you want to look at.
It's mainly doing loads of different comparisons.
Could be, for example, if you have different subgroups or just looking at any kind of structure that the data might be.
What happened to have and then come up and say, this is an answer.
This is the answer to a question. So it is almost really reverse engineering ending up with an answer and then going back and oh,
what was the question that was going to be answered by that? Okay.
Yeah, you know, it's interesting. So but that's a really interesting start.
Let's move on to some of these papers. Okay. What is it about tautology?
The absence of evidence is not evidence of of absence.
And this paper talks a little bit about throwing the term negative in the bin.
And I'll say what is published in the BMJ by Doug Altman and Martin Bland,
which many people will know in the Bland Open Box, which, if you have told me, might come back to that.
So by convention a P-value greater than 5% P greater than 0.05 is called not significant randomised controlled clinical
trial that do not show a significant difference between the treatments being compared are often called negative.
Yeah. Yeah. You know you say I mean this artificial P of 0.05, where does that come from and why do we get in this body?
Do see this? Well, it doesn't work or it's negative.
The threshold, if we want to call it of p point nought five is remnant to when before we have computers.
And before then what had happened is people had looked at specific what we call statistical distributions.
People might know the normal distribution is the main one, which is a bell shaped distribution, but as many other types of distributions,
not only that one, and calculated the specific value for that distribution at different probabilities.
So it would be, for example, a probability of 1%. What would be the value of that distribution?
A probability of 5%, 10%.
And because they calculated these numbers, they became the thresholds that people could then use in the calculations.
So there were just very specific values and the same thing applies for the normal distribution
as to for other distributions are very commonly used in other statistical methods.
For example, the F distribution of the TV screen and maybe what other people have heard of now with the arrival of a computer,
it allows us to calculate the reverse. So for whatever value of X, for example, less than the normal distribution,
what would be the probability that the would be, let's say that the the p value from that number onwards.
And with that approach, it meant that a threshold of oh point nought five became a little bit outdated,
which means that what it boils down to is if you have a certain amount of information,
do your calculation end up with a P value of 0.051 that quantitatively and qualitatively is not different from 0.05.
Hmm. Yeah.
So having a predefined threshold point or five is really a line in the sand which says you'll get it wrong one in 20 times, roughly speaking.
But people are moving away from that threshold and thinking more of what is the actual p value?
Is it very, very small? Is it relatively large and making more if you want qualitative decisions based around that?
Okay. So that's interesting. So one of the things that we often find and people are doing is and this paper is
about is is more around the sort of sample size issues and it talks about here.
Freeman himself found that only 30% of a sample of 71 trials published in the New England Journal of Medicine with a p greater than 0.1,
were large enough to have a 90% chance of detecting even a 50% difference in the effectiveness of the treatment being compared.
Yeah. So when we talk about that, this sort of terminology, alpha and beta isn't the sample size.
So you see this is 80% power to detect a certain difference with an amount of confidence around
that I could use so elaborate tried to explain in a drawing to explain I guess somebody said is it
math with the letters which for us of clinicians is really difficult to understand and just trying
to explain them terms a bit so people could understand how you arrive at sample size calculation.
Okay. So let me give you a brief, some some brief terminology of what we use and set the scene of how we calculate these samples.
Most of the calculations that we do or most of the comparisons that we do are based around medians.
So it means so means of something very good.
Could be, for example, the mean number of individuals, a proportion of individuals, that proportion can be thought of as a similar type of mean.
So the proportion of individuals that recover in one group compared to proportion of.
People that recover if you give them a different type of treatment.
So the proportions can be thought of as a kind of mean if you want other types of means would be,
for example, the mean blood pressure, one division in one group.
So the mean blood pressure, if you don't give them treatment, this is the mean blood pressure.
If you give them treatment. Most of these comparisons around means no means are a really interesting statistic because the more data you have,
the more information you'll have about the overall mean of the population.
And that translates into having more precise information around the mean.
So if we think about that idea of precision as we extend the sample and we gather more precision,
more information about where the actual mean of that group happens to be,
what happens is when you compare two groups, if you have bigger numbers in each one of these groups with increased precision,
you increase the certainty about a separation.
If a separation exists between these two groups, and that's where the term sample size comes in.
The separation that exists within these two groups is a hypothetical separation,
because before we start the experiment, if you want, we don't know that separation exists.
So it's down now to comparing two ideas of two hypotheses.
You might be one idea. First, you usually call it the null hypothesis for the null idea.
There's no difference between the two groups, in which case it doesn't matter how big a sample size is going to be, we're still going to get.
In fact, the the bigger the sample it is,
the more precise we precisely will know that there's no difference between these two things versus another hypothesis,
another idea that there is some degree of separation.
And it is this degree of separation that could be ten millimetres of mercury, for example, or 20 minutes of mercury.
If we're talking about blood pressure, if we have a clear separation, if there's a very big separation, we need less data, smaller sample size.
If there's a smaller separation between these two samples, we need more data because we need to increase the precision.
And thus with this type of alpha, how if we reject,
if we how much information we need to collect in order to decide whether one
hypothesis is correct or we think we have evidence to reject one hypothesis,
in fact or not.
And beta, which is the capacity of us in a given experiment to identify or to collect data to say that this is actually happening is is the.
Yeah, and so what are some people like here they go 90% power and some people say 80% power.
And you put some grant applications in and people say increase the power.
Why why do people do that before?
If we if we think about it in sort of repetitions of the same experiment.
So if you were doing this experiment lots of times because it's probably the easiest way to think about it theoretically,
if you were to do the randomised controlled trial 100 times with these numbers,
let's say you choose if you find that the sample size you need is 200 in each group with those numbers, 200 in each group,
if you write a hundred times a power of 80% of a power,
90% will tell you how many times 80 out of 100 for 90 to 100, you will find a significant again.
Okay, that's how much that's made. That's simple even for me.
An interesting one. Just finish this one point. I think about this paper which published in 1995 in the BMJ.
So it's quite it's not an old paper book if it is, it is new compared to the other two.
Yeah, I compared. And I guess there's something here about probably why systematic reviews came about talks about five fraud analytic treatments,
mainly structure kinase in acute myocardial infarction and the overview of randomised controlled trials,
which is what we now know is a systematic review, found a modest but clinically worthwhile and highly significant reduction in mortality of 22%.
But only five of the 24 trials had shown a statistically significant effect with the p less than four whole point.
Yeah. And I thought with some estimation, I just wonder is this an open source as wider research is on the power or is that still a problem?
We see tonight today that actually many trials are still underpowered and we need to do more.
Where are we actually moving into an age now? Where would you see much larger trials?
I think there's a combination of things that so for one thing, probably nowadays people are more clear on the idea of power.
And if you have a primary outcome, they normally funders would be adamant that you need enough information out there, enough patients,
a large enough sample size to find it best if the if the primary outcome is going to to show significant finding or not.
So the problem of powering on a primary outcome. It's I'm not saying it's solved, but it's not as big as it was maybe 20, 30 years ago.
However, you have to think that each study will be looking at one particular primary outcome
and there might be a whole host of other secondary outcomes that might be. Even more clinically relevant.
But simply because they are less likely to have been reported.
The event rate is smaller. That it means that you need much bigger sample sizes to identify an effect in those types of outcomes.
Which means that a person doing the study may be focusing on a completely and a
proxy which is not as clinically relevant as a secondary outcome by definition,
than it means that that particular study would be underpowered for that secondary outcome.
And that means that you would need still to use all these different studies that look at relatively same populations, have similar questions,
combine them all in systematic reviews to find the real hopefully, if not the real answer,
to gather enough evidence to say, well, we think this is the right answer.
Yeah. Okay, good. And it's interesting. He finishes this. We'll just finish this paper with this sort of bold statement,
which I think is probably were imprinted on your mind when we are told that there is no evidence of A causes B,
we should first ask why the absence of evidence means simply that there is no information at all.
Yes, if there are data, we should look for quantification of the association rather than just a p value quantity.
What does that mean? So I can understand the first, but we haven't got the evidence.
So you shouldn't say this is negative. You should just say, look, we don't know the answer.
You don't know. But this quantification of the association rather than the p value.
So. So that's, that's for example if you have information from randomised controlled trials and it's still uncertain as to what,
what the real answer is because we don't have enough information. You might go back and look at observational studies.
So the classic example might be of serious adverse events where from randomised
controlled trials you might get some idea of what the serious adverse events are,
but you don't have enough information because it's just not enough numbers.
And then you see yourself and see, that's why very, very small.
So you might want to revert back and look at what other sources of information might be
providing you with more relevant data to explore whether there's an association or not.
And that's what that's referring to. So you might want to use other forms of information to quantify if an association might be present.
Okay. Well, that's good. Let's move back now just to 1763.
I notice this small jump. I notice this an essay towards solving a problem in the doctrine of Genesis.
Yes, I notice it was submitted on the 23rd of December three three just in time for Christmas.
Now, this is interesting. I read this.
I actually went over this again yesterday, some bits of a complex, but there's some very interesting bits in there.
But let's I guess the Reverend Mr. Bayes.
Yep. Is somebody who most people coming into statistics, epidemiology will have heard of the concept of Bayesian reasoning.
That's correct. That's in a way the reason why I chose this paper.
Okay, so give me the reason why I chose it and give me, in a sense, if possible, if possible at all, what is Mr. Bailey's issue in Bayesian reasoning?
Well, actually, this paper is not about Bayesian reasoning, which is probably one thing that people might find or.
It's it's not about the usual thing that people associate.
Well, it is. And it isn't the main focus of this paper. It's not about Bayesian reasoning.
It's about trying to solve a slightly different problem, which you can then regard as a basic problem or frequentist problem,
which is the separation between two camps that was went on for many years and probably now it's converging
because people don't necessarily new statisticians or probably not at least in the last decade or so,
use Bayesian or frequentist methods, depending on what's more useful.
Anyway, this paper is the focus of this paper.
It's trying to understand coming up with a series of proofs of propositions to answer the following question a simple,
relatively simple following question. If you have a series of experiments that are independent and you observe the series of experiments,
one of the outcomes of these year's some experiments, let's say tossing a coin, you toss a coin, which is an unfair coin.
So we don't know exactly the probability. If there's a third going, we would know it's 5050 chance.
But it is an unfair coin. So we don't know what the probability of landing heads or tails is, but we observe a series of heads or a series of tails.
And the question that we're trying to answer is if we knew what if we have observed all these outcomes of the experiments?
Can we quantify what are the chances of this probability of heads, let's say, being between a number of another, let's say between 0.8 and .29?
And that's in a way really. Missionary because although he doesn't think about it this way, it can be thought of it in a frequencies way.
It switches around the problem of most of the time, we think.
Given that we know this is a very common 5050 or given that we know that this coin is point three,
the probability of point eight of being had some point of being tails one on one chance of winning eight times.
And this is the other way around. So it switches it saying, okay, given that we observe all this evidence,
what are how can we bound and calculate or determine the probability of heads or the probability of of tails?
Now, within this paper, I think is Proposition four and five,
you find the basis to of what is called Bayesian reasoning, which is all about in the way conditional probability.
So given that one experiment that if you have two events that that only if both events happen,
you will receive a prize and say and you observe either one event and not the other one.
How do you update your probability or calculate the probability on the first event and all
these conditional probability if we then revert that into parameters like for example,
this probability of has or probability of tails gets us back to base and reason directly to Bayesian reason.
So it's interesting. Yeah, I mean, you look at it so it says here,
find out a method by which we might judge concerning the probability that an event has to happen in given circumstances and upon
supposition that we know nothing concerning that under the same circumstances that has happened a certain number of times.
About a certain of the number of times. That's right.
But I guess it goes back to the large pie, because there's a bit here on on page 47 two that the larger the number of experiments,
we have to support a conclusion so much more.
The reason we have to take it for granted. Yes.
And this is at the heart of statistics, if you want.
We can't with statistics, we can't prove anything.
And that's that's the major thing that needs to be ingrained on everyone that
the use of statistics for works in statistics should we can't prove anything.
And it's also sort of part of the scientific method. Science allows for uncertainty,
shoot up and since we think we know but we should always be ready to readjust if the information
that the evidence now that suddenly switches or changes or changed our ideas behind it.
So what here what he's saying is the more information we have, the most certain we should be, which makes sense.
But it doesn't say we should be certain, which is, I think, a clear distinction.
So that's interesting because I'm just looking here through I noted yesterday when I was reading that
there's a particular here on 409 when he talked about when you born and it talks about a second
appearance or one of the return of the sun and an expectation would we might be raised the name of
a second return and he might know that there was another 3 to 1 with some probability of that.
So talked about the fact the sun's going to come up tomorrow and it would increase, but it would never be 100%.
That's correct. Yes. So basically, statisticians can never be certain of anything.
Yes, that's correct. We can never be certain. We can attach a probability to an answer.
I'm 90% sure. Okay. So I think this bit was really interesting to me.
It is lower than this.
Some of this is a bit complicated for me, but this seems to be low even like around normal distributions and everything in here.
And another important thing to highlight of this paper is how much the writing in statistics had changed.
So, I mean, I also have to be honest, I also had trouble following some of his proofs and some of his arguments.
And this is mainly because, although it's using logical argument to present what his proposition for,
his justification for the different propositions that he's he's making in the paper,
it doesn't produce what we now would be the main tool that we now use, which is algebra.
So with algebraic terms, things, people that have studied maths particularly things become a lot more simple to follow.
And the second paper about my now that uses very complex algebra but still
relatively easy to follow compared to the 50 pages or so the Reverend Bias uses.
Well, let's try this out. Well, some of this is on the last page full on a okay Australia.
He says since this was written,
I found a method of considerably improving the approximation in the 2D and 3D rules by demonstrating that the expression and I have.
No chance of seeing that expression.
So I want to test your ability to communicate this equation to our audience today without taking any breath or pauses in between you.
That's that's going to be quite tricky. But let me let me have a go.
So the equation says something like it's a ratio and it's two sigma divided by one plus to
expectation of A to the P to the to the power of p b to the power of Q plus 2e8 to the power of P.
Beat the power of Q the violent. And it sounds it looks very complex.
And the main thing is we use these approximations.
So we use these values many times they come from fairly large calculations,
very large sums that are then simplified with assumptions and arguments that would say, for example, well, is this particular value?
We think it's very, very small so we can get rid of it. Well, this particular way cancels with another thing.
So let's get rid of it and end up with something that will be relatively simple to compute.
Usually at this time it was the 17th,
18th century by hand that would then allow them to use to create an approximation to the calculation that they wanted to obtain.
Nowadays, with computers, we end up with really, really large equations.
So this they talk about an expression that is an approximation. Nowadays we have expressions of many several lines long, which we can then,
including a computer that would get a much more accurate and a value that is much more
closer to the real value that we want to calculate is usually still an approximation,
but it's much closer to what we want. While there's no chance I could say that, so if you can say that without coming up for breath,
that means you're pretty qualified in sophistication and about to be an example.
Let's say just page 376. There's some issues here which I think are definitions which he says are fundamental to probability.
And I guess probability is is a fundamental principle in health care.
And we're always trying to estimate what's the risk of A causing B, but just point one's untreated.
Several events are inconsistent when if one of them happens,
none of the rest can and then point to two events are contrary when one or other of them most and both together cannot happen.
Yes. And I and I, we touched on it in I mentioned it before with these terms.
They're sort of the terms of independent. Independent of in the same way you just explain them because they seem to be important concepts.
Yeah. So, so you have things like but if you have an independent, well this first two terms are mainly about what we call sometimes disjoint events.
So there is a series of events that one of them might happen or not happen.
And in those cases, we, we say that these events are not independent because if one happens, the other will not happen,
for example, and they are particularly useful for defining what are the totality of events that might happen.
And that also defines everything that might might happen if you want.
So that's that's that's a really crucial aspect of probability because then you say, well,
if you have only one type of event death, let's say the country of that would be surviving.
And if you are not dead, your life and that some of those two things as of 2 to 1 now the other the flip side of that is when you have
events that are independent of each other and that normally means that the chance if one of those events occur,
it does not affect the other event. And this is a little more difficult to come up with examples of what these might be.
Let's think about is, for example, whether it's going to be sunny or rainy today,
that there might be one event, let's say it's sunny today for a change here in Oxford.
And then the type of breakfast I'm going to have, I normally have toast, I might have eggs and I haven't seen weather, weather, the weather reported.
And it might be that if it's it might be exactly the same temperature.
So it might not necessarily affect my decision of what I'm going to have for breakfast.
So the probability of me having eggs or having a piece of toast and the probability of it being
sunny or raining or the events are thought to be independent because one will not affect the other.
What tends to happen is that these type of probabilities are not very or independent between two events are not very easy to show in,
particularly in medical terminology,
because that is because many things are associated and the concept of association and correlation then becomes very, very important to use.
Yeah, okay, great. I mean, that's a real education.
This program that we could have probably stayed here for about an hour going over this paper and
there are a whole about ten problems to look out many things in the but let's move on now with.
We're into a paper in the Journal of the Royal Statistical Society,
which you are a member of the Statistical Society with join in if you're interested.
Definitely. Definitely. There's lots of different chapters, medical statistics, primary care.
And I think that's that's there's several that people are interested.
Welcome to join not only statisticians but also people who are interested in statistics, of course.
And then we've got this paper by Jane, Nelda and our W.M. Wedderburn and it's about generalised linear model from 1972.
Yep. So I guess the first thing is to say, well, what's the generalisability in your model?
Mm hmm.
This paper I chose, because it really is, is presenting a series of method that underpins most of the modelling we use or we do in medical statistics.
Not all of it, but most of it. If we think of a model being a way of saying what will happen to, let's say, my health,
given the one of my chances and say of survival of of not having or let's say, of having a cardiovascular event,
heart attack, given that I am a smoker and I'm overweight, I'm an adult 55 that I am in touch with within the next year,
one of my probabilities of having a heart attack in the next year.
That association of the different factors of the individual and something else is a model.
And what these papers are describing is a series,
a type of models that links up pretty much any kind of modelling or most kind of modelling that we do in in medical, medical statistics.
The reason we call linear is because we show each one of these factors would have a linear life if you want to some shape of,
let's say in this case, a transformation of the probability. You might be a lot easier to explain in terms of, let's say, weight and height.
So we want to explain height in relation to weight we can fit aimed at the height.
The taller you are, you're playing along a lot. So y in excess of height against weight.
But what these models do is extend that line,
which would be only in certain circumstances would be useful only when you have a very transformation of X and Y to other types of approaches,
like for example, probability, like for example, risk, like for example, one of the other ones.
So he talks about a plus on distribution and he talks about, well, let me comment on that.
Let me come in on the introduction. And this is the thing for us.
I said simple clinical clinician in the first almost two sentences.
Yeah, let's just listen to these two linear models customarily embody both systematic and
random error component where the errors usually seem to have normal distribution.
The associated analytic technique is least squares.
Very. Yeah.
Now, well, I've got one, two, three systematic random error I need to know about normal distributions to and then I've got these square theory now.
No wonder I'm in trouble. Yes. So then how to give that?
How would you go about sort of thinking this thing through and how would a students then think,
well, you've got to understand these concepts before you can then move on to.
Yeah, these. I think the best way to approach these type of thoughts in this type of paper is to think in terms of,
first of all, this, this what we call the simple linear regression.
So in simple in regression, which is a specialised case and by specialised case we call it a simple case
of this type of models that this paper is presented in superior aggression.
What we have is one viable outcome, whatever you want to call it, that we call it a dependent outcome.
So something that we want to say something about and something that is an independent variable,
that is something that we either have control of or that we know.
So in the in, again, probably the best way to think about it, let's say BMI and blood pressure.
So BMI goes up and see a weight is an it. Yeah,
as a weight go so maybe your blood pressure is also is also going going up and we might want then
to fit align to if we think of available in a graphical shape like a scatterplot as your BMI.
If you plot a BMI individuals with a particular BMI, a particular blood pressure,
you would start seeing a pattern that might follow a line as your BMI goes up, your blood pressure goes down.
And this linear regression is why they. Two is you can just see the line, say, okay, we draw a line on those points.
But because there's going to be small variation in this comes to the idea of either thing you call it
random viability or systematic viability because there's going to be viability between individuals.
And that viability might be because some individuals naturally have lower blood pressure.
So high blood pressure, but they also might be because of the measurement error.
As you take the blood pressure, you might be some error. There's some merit in the measurements.
There's going to be scatter around that particular line.
So we use here where this weighted least squares comes in, we use different methods to come up with the best possible line.
And that best possible line in the case of a simple linear regression is one that minimises the difference.
We'll go from each point to the line, and that's what the least squares comes in an end with this.
And then the coach, I'm trying to understand these folks so we've got weight and say blood pressure going up in a line,
but some people will vary from that line. Right.
And the difference in their variation, you'll use the least squares method to come up with the best possible line.
And there are two sources of variability, the random variability that occurs in blood pressure.
But there may be some systematic variability because the blood pressure curves not working, for example.
And then look, this is if I can get this, I'll be happy.
Then obtain the approximation to the weights isn't there will be to the reward.
So what? So you see the what if we can just get that that would not be the panacea if that light.
So you have an A the A line which is almost a regression line and then you talk about either the beaches or the weights of each component.
There's like two things. So the weights, the weights that they talk about in terms of of weight, at least squares,
is in relation to how much weight you're going to give to each one of these points.
And if you are talking of simple linear regression, you give equal weights to each one of the points.
If, for example, you're talking about something slightly more complex where you think some points have more information than others,
you give more value to those points. Similar to what happens in meta analysis, where you might give more weight to studies are bigger.
That's one area the betas are you talking about.
It's more of the quantification of how much things change by a change of one unit in your independent variable.
Let me be clear when we're talking about this line, if you think of a line, the line would have a which is completely defined by an intercept.
That is the point where if you had zero BMI, this is completely theoretical,
but if you had a similar BMI, what would be your blood pressure say might be 20.
Let's a hypothetical. So that's that's that in a way uncross your line tells you where you start.
And then the other bit that completely defines your line is the slope.
So by a change of one unit and BMI, how much your blood pressure is going to change.
And it might be that when you move from 20 for a BMI of 25 to a BMI of 26 to blood pressure increases by two millimetres millimetres
of make it so that to the change in one unit in one unit of BMI is an increase of two units of systolic blood pressure.
That two is exactly your slope. Okay, now the slope and the betas are exactly the same thing.
Okay. So when you want to do with a combination of lean squares or whatever you want to call it, you know,
to determine your betas is basically to determine the slope or the increase in the
dependent variable that you obtain by increasing your independent by one unit.
So that so basically you, like we said, you may have blood pressure, may be affected by weight, you may add another factor in like age.
Yeah. Which would come into your model. That's right. And that would increase the slope or decrease it.
What happens is that each one of these characteristics have their own slopes.
Yeah. So BMI might have a slope of two, but once you bring age, your slope of the of BMI might decrease, as you were saying to one or 1.2.
And there is another slope of three or 1.2 as well.
And so then the only other factor, if we get these folks, we're running out of time is this this sort of idea in models is how much can be
explained by the model and how much count what that value and how to test for that.
So we're talking about the scatter that you obtain. Yes.
And because there's going to be scatter, I mean, it could be natural variability or whatever it is,
that scatter is precisely that you can't explain through your model.
Yeah. So your line is brilliant if you only have two dots.
Yeah. So you can explain everything.
If you have that's once you have three dots or more, three individuals or more, there's going to be some variability you won't be able to explain.
And that very belief that you won't be able to explain tells you how good your model is.
Okay.
And there's different ways of quantifying how good your model is in terms of how much of the total variability of the data you manage to explain.
Okay.
And that could be in terms of, for example, just looking at the differences from design to each one of these dots or other types of measurements.
Gosh. So that's been an amazing, very quick journey.
40 minutes on probability sample five linear models.
Now let's just finish with, let's say, a bit of advice.
What would you say the best advice if you're embarking on a course epidemiology statistics?
What sort of resource for what would be the best way to just keep improving your statistical knowledge, improving existing knowledge?
I think what has worked for me is getting your hands dirty.
So doing the things,
having some idea of what you want to achieve or what you can possibly achieve through either lectures or through going to courses.
But more than anything is the in applying these things.
So if, for example, you have data and you want to use a particular model that has a linear model of these claims, then go ahead and do it.
Ask for advice. Many times there's advice either in the statistical packages,
also in the in the in the text and also in in the individuals in your own own organisation or elsewhere.
And the earlier you go for advice, the better that will be. The other thing that I would I would say.
Okay. Well, on that note, let me just let me just finish here.
You go. And you got to answer one of the two. Okay. Mean all the median?
Medium. The like a true statistician has to think about it.
Stator or are are all the time.
Okay. There you go. I going with speed. You know, while on that now I'd like to say thank you very much to Rafael Perera for what's
been a very interesting discussion on statistics and many issues around that.
Thank you. You're welcome. Thanks. You're welcome.