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0:00hello and welcome to the next video in
0:02my series on basic statistics if you are
0:05a firsttime viewer please stick around
0:07for the intro it is worth the time at
0:09least once if you are a regular viewer
0:11feel free to skip ahead using The
0:13annotation so first a few things I do
0:16these videos because I love to learn and
0:18help others learn we are all good at
0:20something so I encourage you to give
0:23back to the world in a similar way share
0:25your passion whatever it is any way you
0:28can now this video focuses on basic
0:31stats and is not a quick fix it aims to
0:34be thorough my goal is an understanding
0:37of fundamental concepts and that takes
0:39time but when you understand the
0:40fundamentals learning other topics is
0:43much easier now related to that if
0:46you're watching because you are
0:47struggling in a class or at work I want
0:50you to stay positive and keep your head
0:51up you can learn this I have faith in
0:55you many other people around you have
0:57faith in you and so should you
1:00feel free to connect with me on LinkedIn
1:03go+ Twitter and of course subscribe here
1:06on YouTube now if you think there is
1:08something I can do better please leave a
1:10constructive comment below I do take
1:13those comments into account when I make
1:14new videos I also encourage you to talk
1:17with other viewers in the comments help
1:19each other when you can and finally if
1:22you like the video please give it a
1:24thumbs up share it with classmates or
1:27colleagues and put it on a playlist for
1:29review later that does encourage me to
1:31keep making them for you so let's go
1:34ahead and start
1:37learning so this video is the next in
1:40our series about multiple regression now
1:42it's really an extension of part four
1:44and in part four we looked at the case
1:46where we had one dummy variable in this
1:49video we're going to extend that to two
1:51dummy variables we will see how to set
1:53up the problem we will see how to
1:54conduct the analysis and then interpret
1:56our result in the end now if you did not
1:59watch part 4
2:00I highly recommend going back and
2:01watching it before proceeding with this
2:04one so that small warning aside let's go
2:07ahead and get
2:10started so as in the last video you are
2:13an analyst for a small company that
2:15develops house pricing models for
2:17independent Realtors to generate your
2:19models you use publicly available data
2:22such as the list price the square
2:23footage the number of bedrooms number of
2:25bathrooms among other data points that
2:27you
2:28collect now now in this problem you're
2:30interested in two main questions number
2:33one is the public high school in the
2:36neighborhood exemplary which is the
2:38highest rating and how is that rating
2:41related to the home price if at all
2:44number two what region so north south
2:47east or west of the city is the home
2:50located and how is that related to the
2:52home price if at all now you'll notice
2:55that this data is not quantitative it is
2:58qualitative they are categ atories in
3:01the first case the high school is either
3:02exemplary or it's not it's a yes or no
3:05question in the second case it's just
3:07the region or the directions of north
3:09south east or west so because we have
3:12categorical data we will need to use
3:14dummy variables in our regression to
3:16conduct this
3:19analysis so in this problem we are going
3:22to use 100 fictitious observations I did
3:26make up this data for this video so here
3:29are the first 15 observations let's go
3:31ahead and look at our data structure in
3:33the First Column we have our dependent
3:35variable which is price so the first
3:37home is
3:41$450,000 now in the second column we
3:43have our first independent variable
3:45that's the square footage so that first
3:47home is
3:493,860 square ft it's a very big home in
3:53the third column we have our second
3:55independent variable and that is whether
3:57or not the high school is exemplary or
3:59not so it's either yes or no and then in
4:03the fourth column we have our third
4:05independent variable which is region so
4:08north south east or west and that's how
4:11our data was originally collected now
4:13the first thing we're going to do is
4:15some recoding we're going to change our
4:17text entries such as yes no north south
4:20east or west into numerical values and
4:22we can do that like this so for examplar
4:26high school if it's a yes we code that
4:28as one if it's it's a no we code that as
4:30zero now for location we do it a bit
4:33differently I'm going to code these in
4:35alphabetical order and I do that to make
4:37many tab happy which I'll talk about
4:39later so East is the first alphabetical
4:42region so we'll give that a code of zero
4:45and then we have North which is a code
4:46of one then we have South which is a
4:49code of two and finally we have West
4:52which is a code of three so up to this
4:54point we've gone from text entries to
4:57numerical entries but we have not
4:59created our dummy variables yet we're
5:01going to go ahead and do that
5:02now so here we have what I call
5:05dumbification so the price column is the
5:08same the square footage column is the
5:10same the exemplary High School column is
5:12the same but now we have three dummy
5:15variables South West and North now you
5:19might be saying to yourself wait a
5:20minute we have four locations north
5:23south east and west but we only have
5:25three dummy variables well if you
5:27remember from part four our number of
5:29dummy variables is always the number of
5:31categories minus one so we have four
5:34categories north south east west minus
5:37one is three so we have three dummy
5:41variables now I think it's pretty
5:42straightforward to see how these are
5:44coded so for the first home we have it
5:47as being in the South Region so over
5:50here on the right for South we put a one
5:52under the South column now for the
5:54second home it's in the North Region so
5:57in that case we put a one in the the
5:59north column and the other regions are
6:01zero so you can kind of see how this
6:03works but you'll notice that there's no
6:06column for
6:08East Now by far the most important thing
6:11the most valuable thing you can do
6:12before you ever run a test before you
6:14ever crunch the numbers or anything is
6:17to look at some Scatter Plots of your
6:19data so here we have the price of the
6:22home versus its square footage along the
6:24x axis at the bottom we have the square
6:26footage and on the Y AIS on the left we
6:29have the price now we can see that this
6:31is a pretty linear relationship if we
6:34just sort of eyeballed a best fit line
6:36we would see that it runs sort of
6:37through the middle of the data now I
6:39would argue that this is probably a
6:40curval linear relationship but in this
6:43case we are doing linear regression
6:45we'll talk about nonlinear regression at
6:47a later point but we can see that this
6:49is a definite pattern of a smaller home
6:52having a smaller price and larger homes
6:54having larger
6:58prices now this Scutter plot is a bit
7:00more interesting as it separates out our
7:02data into groups so as you can see the
7:05blue dots represent homes where the high
7:08school is not exemplary and the red
7:11squares represent homes where the high
7:14school is exemplary now you can see a
7:17definite pattern here look at the blue
7:19dots where are they well they're
7:22definitely along the bottom of this
7:24scatter plot so especially if you look
7:27from about 2,000t homes up to 3,000 ft
7:31homes the blue dots are consistently
7:34below the red squares now we can go
7:36ahead and put some lines just sort to
7:38eyeball it and there they are so let's
7:40step back and look at the big picture
7:43let's take a average home that is
7:462500 ft now we can see that for a home
7:50that's 2500 ft the price of a home where
7:52the high school is not exemplary the
7:55blue dots in line is about $150,000
7:58[Music]
8:00now for the same size home 2500 sare ft
8:04if the high school is exemplary the home
8:08is a little bit over
8:11$200,000 so for the same size home the
8:15only difference being whether the high
8:16school is exemplary or not we have a
8:19price difference of over
8:23$50,000 so we can think of that as the
8:25exemplary High School premium price
8:29given the same size home the one where
8:32the high school is exemplary Demands a
8:35price that is about $50,000 more and I
8:38find things like that very interesting
8:40so let's go ahead and keep that in the
8:41back of our minds as we go
8:46forward so here we have a very busy
8:48graph here we're looking at the regions
8:51on the same basic scatter plot so we
8:53have East which are blue dots North
8:56which are red squares South which are
8:58green diamonds and West which is purple
9:01triangles so what general patterns do we
9:03see here what looks like the West holes
9:06the purple triangles stay along the
9:08bottom of the scatter plot the north
9:10homes kind of stay along the top of the
9:12scatter plot and then you know the other
9:15ones the East and the South kind of stay
9:19in the middle of the scatter plot so
9:21let's help ourselves out by putting some
9:23best fit lines on this
9:27graph now that's a bit easier to read so
9:30we can learn a ton of information just
9:32from this scatter plot so let's look at
9:34the West homes first so those are the
9:37purple triangles as we can see they're
9:39along the bottom of the scatter plot
9:42they start at around,
9:441500t the largest one is about 3500 ft
9:48now the price does not increase as
9:52steeply as the other regions you can see
9:55that the slope of the purple line of the
9:57dash line is quite a bit shallower than
10:01the slope of the other three lines so in
10:04the west region as the homes get larger
10:08the price doesn't go up as much as the
10:11other regions now let's look at the
10:13green the South Region so we have the
10:16green diamonds now there the smallest
10:18home is about 2,000 sare ft and they go
10:21all the way up to almost 4,000 sare ft
10:24now look at the slope of that line it is
10:28Extreme L steep so as the homes get
10:32bigger the price shoots up very rapidly
10:35as compared to the other regions so we
10:37can sort of see the difference here in
10:40the west we have smaller homes that are
10:43less expensive the price increases less
10:46as the home gets bigger in the opposite
10:48extreme we have the south region where
10:51the homes start a bit bigger and as they
10:53get bigger the price goes up
10:55dramatically and then we have the other
10:57two the East and the North which kind of
11:00hang around in the middle so we can
11:02definitely see the relationship between
11:04square footage the size of the home and
11:06the price as separated out by region so
11:10keep that in mind as we go
11:14forward so here we're going to look at a
11:16surface plot now the purist out there
11:19I'm going to note because I know you're
11:21going to say something probably that
11:22this is not the best use of a surface
11:25plot and I know that I'm just using this
11:28as a visual tool to show the difference
11:31in price in square footage across
11:34another variable so here we have a
11:36surface plot of price which a square
11:37foot in three dimensions and exemplary
11:40high school you can see that we have the
11:42price and the vertical axis we have the
11:44square footage in the z-axis front to
11:47back then we have the exemplary High
11:49School category running along the x-axis
11:52here in the front here's the question is
11:56the surface tilted downward where where
11:59the exemplary High School equals zero so
12:01down here in the lower left corner is
12:03this sort of tilted down into the
12:06left so if we're looking at this from
12:08the front view if we were standing right
12:11where it says exempt High School exempt
12:14HS down here at the bottom if we stood
12:16there and looked at it would it look
12:18like this would it start at the bottom
12:20and then kind of slope up up towards
12:22where we have one well I would say yes
12:27so what does that tell us that tells us
12:29that as we go from 0 to one from a not
12:33exemplary High School to an exemplary
12:36High School the price and the square
12:39footage both tend to increase so this
12:42threedimensional plane is sort of tilted
12:45down here at
12:460.0 and kind of goes up as we go to the
12:49right and somewhat to the back and
12:52that's how we can use this 3D plot to
12:54visualize our data and again it's not
12:56the proper use of a 3D plot but it does
12:58help us visualize this in three
13:02dimensions so here's a surface plot of
13:04price and square foot again but this
13:06time location is here in the front axis
13:10so what are we looking for here now are
13:13there front to back what I would call
13:15sort of mountain ranges at any location
13:18so do they run front to back like this
13:22does any location have higher prices
13:26across the surface taking into account
13:28the square footage well what do you
13:30think about that well I think they all
13:34tend to follow the same basic pattern or
13:36most of them do we can definitely tell
13:38that the east region front to back has
13:41that one Spike at around 3,000 square ft
13:44the South kind of goes up as it proceeds
13:47to the back the west region however
13:50really increases in price and square
13:53footage as it goes to towards the back
13:55and the north maybe a little bit less so
13:58so here we can see that the west region
13:59it seems like has the biggest change in
14:03square footage and price as we increase
14:06both of those
14:10Dimensions okay so this is probably the
14:12most complicated slide in this uh
14:14presentation so I want to do it real
14:15slow so you understand where everything
14:16is coming
14:17from so here is our estimated regression
14:20equation so we have the expected value
14:22of y which is our dependent variable
14:25equals beta Sub 0 which is our intercept
14:28and then we have our series of
14:30independent variables from beta 1 X1 all
14:33the way up to Beta 5 X5 now remember
14:36these all stand for something so we have
14:38our constant which is our intercept and
14:40then beta 1 X1 is our coefficient and
14:43then X1 is our square footage variable
14:46then X2 is our exemplary High School
14:48variable and then Southwest and North
14:51are our dummy variables we can think of
14:53those kind of as a group so here are all
14:56of our variables as represented in our
14:58estimated regression
14:59equation now let's look at an example
15:02what's the expected value of the home
15:04price given that the high school is not
15:08exemplary so in that case X2 is zero if
15:14you look up here for exemplary high
15:16school at the top that's our X2 variable
15:19so if it's not exemplary we know that
15:21that value is a zero and the home is in
15:25the west so remember for X4 that's our
15:29West dummy variable so that would be a
15:31one in that spot so we can write this
15:33out using some notation so the expected
15:36value of y our dependent variable given
15:38that's sort of the vertical line pipe
15:40there that is not exemplary and it's in
15:43the west equals beta 0 plus beta 1 X1
15:47plus remember that's our square footage
15:49plus beta 2 X2 but in this case X2 is
15:54zero because the high school is not
15:57exemplary then we have beta 3 0 well why
16:01is that that's because the home is not
16:03in the south then we have beta 4 * 1
16:07well why is that a one that's because
16:09our home is in the west if we look up
16:11there we have beta 4 X4 where the West
16:14is so that's a one then we have beta 5 0
16:18that's a zero well why because the home
16:20is not in the North so what we can do is
16:23actually do some simple algebra here and
16:25reduce this down to this so we have beta
16:290 + beta 1 X1 + beta 4 now where does
16:33that come from well if you look at the
16:35one above it anything that's multiplied
16:37by zero goes away so beta 2 goes away
16:40beta 3 goes away and beta 5 goes away
16:43because they're all multiplied by 0 so
16:46we're left with beta 0 + beta 1 X1 +
16:50beta 4 because beta 4 * 1 is beta 4
16:53itself so that is our simplified
16:56estimated regression equation for this
16:58particular
16:59example so what are the total number of
17:02possible equations we could actually
17:04have in this problem well we're always
17:08going to have one constant which is our
17:11beta 0 times our 1 s foot variable
17:15that's just a simple quantitative
17:17variable times two for our exemplary
17:19High School variable it can take the
17:22value of a one or zero so in that spot
17:26we have two possibilities it's either
17:28one or zero now for the region we have
17:32four possibilities so in this case we
17:36could have 1 0 0 which is South we could
17:39have 0 1 0 which is West we could have
17:41001 which is north or we could have 0000
17:450 which is a home in the east region so
17:50we multiply all these possibilities for
17:52each part of our estimated regression
17:54equation we have eight possible
17:58equations
17:59in this multiple progression model with
18:01these two dummy variables so 1 * 1 is 1
18:04* 2 for two possibilities for examplar
18:07high school times four possibilities for
18:10the region which we can see over here on
18:12the right 1 * 1 * 2 * 4 is 8 and we'll
18:16actually see all eight of these
18:18equations here in a minute or two

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