Fact Check: Linear Regression in Ultrasound Fetal Weight Estimation Statistics

so as promised we are going to talk about linear regression briefly

um we're actually going to spend an entire chapter talking about linear regression

so this is really just an uh a very slow introduction smattering of the ideas

but here i have an example a real example actually um it's a scatter plot so each

blue diamond is a sample data value of ultrasound fetal weight estimation so basically when an

ultrasound tech estimates the weight of a fetus in the in the uterus while the the baby is

growing and then the actual birth weight of the child so once the baby is born they measure it

using a more accurate procedure but the idea then is we have this scatter plot we have a good

idea that there might be a linear correlation trend here

now i want to call your attention to this piece right here so often when you encounter um a

scatter plot with a regression line they're going to give you the r squared value

and just to remind you so what we what they call r squared right is also known as the

coefficient of determination it's exact meaning we'll talk about

later in this course but what's important about it is that we get the coefficient of determination from

the linear correlation coefficient so i mean literally r squared is r squared

which means conversely if you have the r squared value you can find what r is actually one really important thing

to note is that r squared is always positive okay so it's a positive statistic about

the strength of correlation it doesn't tell us the the direction unless we have the scatter plot

then we know the direction of the the correlation and here we can see that it's positive because

the trend line has a positive slope okay so then we could you know calculate what r would have to be so if

you calculate the square root of 0.7646

7646 one interesting thing to note and you may remember this from from algebra that if your your

number that you're taking the square root of if it's between 0 and 1 that means the square root is actually

going to be greater so it's counterintuitive it's interesting which means that r

here is 0.8 roughly okay and we know it's positive again

because we have positive linear correlation here and this is this is evidence that you

know we have a strong linear correlation here okay so we're justified in setting the regression line

so what the regression line is right so it's a model it's a mathematical model or actually i

should say statistical model so the regression line has a form y hat equals

a x plus b is the equation of a line okay and y hat is the predicted

y value that corresponds to the x value that we pick okay in other words in this in the

context of this example let's say we're thinking about a new data value okay so an ultrasound

just came through into the database and we found a birth weight or sorry an ultrasound

weight of 300 3500 grams this is all in grams right so it's it's not in pounds it's you know

um so based on that we can use a regression line to predict what the actual birth weight would be

specifically it's a point on that line right and the the actual birth weight right you can figure out from the line

so in other words our 3 500 grams is our x value that we would substitute

in and if we had the equation of this particular regression line

then we could figure out um what exactly we're looking at here does that make sense

now since we don't have the equation then we can't we can't do this precisely okay but in

general this predicted y value if we have a a if we have a data point somewhere in the

sample that doesn't match that exactly that happens that's okay in other words the prediction might not always bear out

okay but the regression line is like our best estimate and we also call this the line of best

fit where in some way which we'll discuss later this is kind of the best

approximation or the best prediction that we have for finding the y value for an x value

for bivariate data okay um and so regression lines are actually super

powerful okay we use regression in a lot of different contexts

this is what is also sometimes called simple linear regression because there are some simple

assumptions that we take here and as we'll see in a later chapter there are more

complicated forms of linear regression including what we call multiple regression which is when we

have multiple variables okay so no longer are we talking about bivariate data

but we would be looking at multivariate data with three or four or five or six however many components

okay but this is characteristic this is the regression line template and some things to to note here

just a review in case you're um you know in case you're a little rusty on your algebra totally understand that

happens everyone including myself so b right b is what happens when your x value's zero

so we can call that the uh base rate or baseline

response value so what happens when x equals zero right so what happens when we don't have

any in other words what if the ultrasound fetal weight is zero right which isn't

it doesn't make sense in this context but if it's 0 then what would b be that's

what we're talking about here and a is the slope of this line in other words it is

how fast does the y value change with respect to the x value so we call that

the rate of change in y with respect to x it's a slope right so if a is

positive then we know that we have positive linear correlation

and if it's negative then we have negative linear correlation