1
00:00:01,000 --> 00:00:03,700
And finally interpreting coefficients.

2
00:00:03,700 --> 00:00:06,233
In front of us here,
we've got the four models that we built

3
00:00:06,233 --> 00:00:09,233
in our backward elimination method.

4
00:00:09,266 --> 00:00:13,500
And as we agreed, model number three is
the best one is the best fitted model.

5
00:00:13,766 --> 00:00:16,766
And if we were to deliver a model
on our project it would be this one.

6
00:00:17,100 --> 00:00:20,100
So let's go ahead and start interpreting
coefficients by looking at this model.

7
00:00:20,666 --> 00:00:24,166
Here we've got two variables
R&D spend and marketing spend.

8
00:00:24,166 --> 00:00:25,600
So it's basically how much

9
00:00:25,600 --> 00:00:29,233
companies are spending on their research
and development and on their marketing.

10
00:00:29,700 --> 00:00:32,766
And we've also got the constant
and we're predicting profit.

11
00:00:33,400 --> 00:00:38,466
So these coefficients
here these are B1 and B2.

12
00:00:39,533 --> 00:00:40,566
What they're telling

13
00:00:40,566 --> 00:00:44,066
us is that so
how to how to interpret them.

14
00:00:44,333 --> 00:00:45,600
First of all you look at the sign.

15
00:00:45,600 --> 00:00:50,866
If the sign is positive
that means your variable is correlated

16
00:00:51,366 --> 00:00:56,100
with your your independent variable is
correlated with your dependent variable,

17
00:00:56,733 --> 00:01:01,766
meaning that if you change,
the value of your independent variable,

18
00:01:02,000 --> 00:01:05,266
then the value, then you can see that

19
00:01:05,266 --> 00:01:09,266
the dependent variable
will be changing in the same direction.

20
00:01:09,300 --> 00:01:12,333
So basically if you'll be increasing
spend on R&D then

21
00:01:12,333 --> 00:01:15,500
your profit will be increasing
your increasing spend on marketing.

22
00:01:15,500 --> 00:01:17,966
Then your profit is also increasing
which makes sense right.

23
00:01:17,966 --> 00:01:21,200
So if you're spending more on research
and development and making your product

24
00:01:21,200 --> 00:01:26,200
better, then, probably your profits
should increase eventually as well.

25
00:01:26,400 --> 00:01:27,366
Same thing with marketing.

26
00:01:27,366 --> 00:01:30,666
The more you spend marketing,
the more you sell, and therefore

27
00:01:30,933 --> 00:01:32,166
the profits should also go up.

28
00:01:33,400 --> 00:01:35,100
And that's the sign.

29
00:01:35,100 --> 00:01:37,200
If the sign is negative,
if the sign were negative,

30
00:01:37,200 --> 00:01:38,400
then it's the opposite effect.

31
00:01:38,400 --> 00:01:38,900
So basically

32
00:01:38,900 --> 00:01:42,433
you increase your independent variable
and your dependent variable decreases.

33
00:01:43,733 --> 00:01:45,233
And now let's look at the magnitude.

34
00:01:45,233 --> 00:01:50,200
So here you can see right away that
the magnitude is higher for indie spend.

35
00:01:50,200 --> 00:01:53,100
And the magnitude is low for marketing
spend.

36
00:01:53,100 --> 00:01:56,033
Magnitude is always tricky
with regressions.

37
00:01:56,033 --> 00:01:57,533
Be careful with magnitude.

38
00:01:57,533 --> 00:01:59,666
Sign is kind of, you know, pretty obvious.

39
00:01:59,666 --> 00:02:04,100
It's either it's one way or the other
magnitude can, really trip you up.

40
00:02:04,100 --> 00:02:05,233
And I can give you an example here.

41
00:02:05,233 --> 00:02:08,933
So you might think that okay, right away
magnitude is greater.

42
00:02:08,933 --> 00:02:13,166
So this this, coefficient for R&D
spend is bigger than the marketing

43
00:02:13,166 --> 00:02:13,833
spend coefficient.

44
00:02:13,833 --> 00:02:16,833
So definitely R&D
spend has a bigger impact.

45
00:02:16,833 --> 00:02:21,166
Well, what if I tell you
that I could easily without changing

46
00:02:21,166 --> 00:02:25,000
anything in the regression, I could easily
make this coefficient bigger,

47
00:02:26,033 --> 00:02:26,866
by a lot,

48
00:02:26,866 --> 00:02:29,866
by a thousand times bigger than R&D spend.

49
00:02:30,000 --> 00:02:31,233
Well, it's easy to do.

50
00:02:31,233 --> 00:02:35,300
All I have to do is say marketing spend
instead of looking at it in dollars.

51
00:02:35,433 --> 00:02:38,233
How about I look at it
in, a fraction of a cent?

52
00:02:38,233 --> 00:02:40,600
So, every marketing.

53
00:02:40,600 --> 00:02:43,833
So let's look at and sense
so marketing spend instead of

54
00:02:43,833 --> 00:02:48,066
counting it in dollars, let's count it in
cents right away because our variable

55
00:02:48,066 --> 00:02:52,100
is going down 100 times, our coefficient
will proportionally increase.

56
00:02:52,100 --> 00:02:53,300
So it go up 100 times.

57
00:02:53,300 --> 00:02:57,266
And if I, change my data
and replace everywhere

58
00:02:57,266 --> 00:03:01,000
marketing spend with sense and everything
else, I will leave the same.

59
00:03:01,266 --> 00:03:02,633
Then I rerun this model.

60
00:03:02,633 --> 00:03:07,966
I guarantee you that the coefficient here
will actually become 2.99.

61
00:03:07,966 --> 00:03:09,566
So you'll just increase 100 times.

62
00:03:09,566 --> 00:03:11,233
Everything else will stay the same.

63
00:03:11,233 --> 00:03:14,300
And that way right away you'll see
oh well the marketing spend

64
00:03:14,300 --> 00:03:16,300
coefficient is greater.
So then marketing spend

65
00:03:16,300 --> 00:03:21,033
has a bigger impact
on, your dependent variable profit.

66
00:03:21,033 --> 00:03:23,766
And that's a mistake
that a lot of beginners make.

67
00:03:23,766 --> 00:03:26,100
You should not fall into this trap.

68
00:03:26,100 --> 00:03:27,533
magnitude is a tricky thing.

69
00:03:27,533 --> 00:03:30,533
And the way to think about
it is always say

70
00:03:30,900 --> 00:03:35,233
magnitude in terms of units
of the independent variable.

71
00:03:35,233 --> 00:03:39,933
So the correct way to analyze this is even
without knowing what they're measured.

72
00:03:39,933 --> 00:03:41,766
And maybe this one's measured
in thousands of dollars

73
00:03:41,766 --> 00:03:44,133
and this one's measured in dollars
I don't know.

74
00:03:44,133 --> 00:03:47,666
For instance, although I do
I do know of course, I've seen the data,

75
00:03:47,666 --> 00:03:51,700
but if, say, I don't know, the
you can still make a conclusion,

76
00:03:51,900 --> 00:03:55,800
just all you have to say is, R&D spend

77
00:03:56,266 --> 00:03:59,266
has a greater impact on profit

78
00:03:59,866 --> 00:04:02,866
per unit of R&D spend,

79
00:04:03,100 --> 00:04:07,633
then marketing, then marketing spend
has per unit of marketing spend.

80
00:04:07,933 --> 00:04:08,533
And that's all.

81
00:04:08,533 --> 00:04:12,433
So basically, even if they're measured
in different things, by saying per

82
00:04:12,433 --> 00:04:17,366
unit of the underlying variable,
you are protecting yourself

83
00:04:17,366 --> 00:04:22,233
from that error that you know they are 
they are measured in different things.

84
00:04:22,800 --> 00:04:25,266
And moreover, imagine if they're measured,

85
00:04:25,266 --> 00:04:27,933
one's measured in dollars,
the other one's measured in,

86
00:04:27,933 --> 00:04:31,733
you know, kilometers or something that
you can't compare dollars and kilometers.

87
00:04:31,733 --> 00:04:31,933
Right.

88
00:04:31,933 --> 00:04:36,433
But you can always say per unit
and this leads us to the actual

89
00:04:36,433 --> 00:04:40,966
interpretation of these variables
or these coefficients.

90
00:04:41,200 --> 00:04:42,600
What does it mean? So

91
00:04:43,766 --> 00:04:47,400
this means 0.79 means that for every unit,

92
00:04:47,633 --> 00:04:51,233
if you keep all other variables constant,
so you only have one other variable.

93
00:04:51,233 --> 00:04:56,333
So if you keep everything else constant
but you are able to adjust R&D spend for,

94
00:04:57,033 --> 00:05:00,033
in this model
or for a, for a hypothetical company,

95
00:05:00,200 --> 00:05:04,166
for every dollar
or for every unit of R&D spend that you,

96
00:05:04,766 --> 00:05:07,766
increase your profit will increase.

97
00:05:07,866 --> 00:05:11,166
According to this model,
your profit will increase by $0.79.

98
00:05:11,866 --> 00:05:15,233
That's exactly
what this, coefficient is saying.

99
00:05:15,266 --> 00:05:18,633
And so for every unit
that you decrease in your R&D spend,

100
00:05:18,766 --> 00:05:24,233
your profit will decrease by 70.79
of a unit of profit.

101
00:05:24,900 --> 00:05:28,233
And because R&D spending, mission dollars
and profit is also million dollar,

102
00:05:28,500 --> 00:05:34,133
that means so for every unit increase
in for every dollar

103
00:05:34,166 --> 00:05:38,733
increase in R&D spend,
your profit will increase in, by $0.79.

104
00:05:39,700 --> 00:05:41,866
So let me just repeat that again

105
00:05:41,866 --> 00:05:46,733
that you are looking at units
increase in R&D spend.

106
00:05:46,866 --> 00:05:51,433
They translate through this coefficient
into units increase in profit.

107
00:05:51,600 --> 00:05:54,600
So if your profit was measured in apples

108
00:05:55,100 --> 00:05:59,933
then a $1 increase and one unit increase
is always going to be true.

109
00:05:59,933 --> 00:06:03,533
One unit increase in R&D spend will drive

110
00:06:03,800 --> 00:06:07,266
point 79 units increase in profit.

111
00:06:07,266 --> 00:06:09,833
And of of course, of the profit units.

112
00:06:09,833 --> 00:06:12,466
So let's turn that into dolls and apples.

113
00:06:12,466 --> 00:06:16,100
A $1 increase in R&D spend

114
00:06:16,100 --> 00:06:21,866
will drive a 0.79,
or 80% of an Apple increase in profits.

115
00:06:21,866 --> 00:06:26,200
So here you're talking about units of R&D
spend in profit.

116
00:06:26,200 --> 00:06:28,500
You're talking about units of profit.

117
00:06:28,500 --> 00:06:31,500
And this coefficient,
it links them together.

118
00:06:31,533 --> 00:06:34,533
So as long as you say per unit
you're fine.

119
00:06:34,566 --> 00:06:35,666
Once you already know

120
00:06:35,666 --> 00:06:39,433
what these variables are measured
and then you can start comparing them,

121
00:06:39,433 --> 00:06:40,566
if they're on the same scale.

122
00:06:40,566 --> 00:06:42,200
And in this case
they are on the same scale

123
00:06:42,200 --> 00:06:44,433
because they're measured,
everything's measured dollars here.

124
00:06:44,433 --> 00:06:47,800
And you can say that, a dollar increase

125
00:06:47,800 --> 00:06:53,800
in R&D spent drives $0.79, dollars
$0.79 increase in profit and mark a dollar

126
00:06:53,800 --> 00:06:58,200
increase in marketing spend drives only,
$0.03 increase in profit.

127
00:06:58,200 --> 00:07:01,300
So basically,
if you're the venture capitalist,

128
00:07:01,566 --> 00:07:03,866
which companies are going to invest
in, well,

129
00:07:03,866 --> 00:07:06,766
from this model you're going to decide
that you should invest into

130
00:07:06,766 --> 00:07:09,766
companies are spending more on their R&D.

131
00:07:10,233 --> 00:07:12,400
there may be lots of reasons behind this.

132
00:07:12,400 --> 00:07:14,400
And this is not just a random fact.

133
00:07:14,400 --> 00:07:18,733
This could be true in reality
because profit is revenue minus expenses.

134
00:07:18,733 --> 00:07:22,166
Maybe marketing does drive
a lot of revenue to a company,

135
00:07:22,166 --> 00:07:24,900
but at the same time,
maybe for these specific companies

136
00:07:24,900 --> 00:07:28,866
that we're looking at their expense,
the cost of marketing.

137
00:07:28,866 --> 00:07:33,600
So, the prices that they pay for
marketing is so high that the increase

138
00:07:33,600 --> 00:07:37,233
in profit, the net increase in profit
is actually marginal.

139
00:07:37,233 --> 00:07:40,333
So the marketing is eating up
a lot of the revenue that it's creating,

140
00:07:40,333 --> 00:07:43,333
whereas R&D is, 
you know, it's creating a lot of,

141
00:07:43,733 --> 00:07:46,466
revenue and a lot of it is actually
staying in profit, something like that.

142
00:07:46,466 --> 00:07:48,400
But we're not worried about that
right now.

143
00:07:48,400 --> 00:07:49,400
That's more financials.

144
00:07:49,400 --> 00:07:52,200
What we're doing
is we're delivering a model.

145
00:07:52,200 --> 00:07:55,866
So that's how you interpret coefficients
with, linear regression.

146
00:07:55,866 --> 00:07:57,300
It's very simple.

147
00:07:57,300 --> 00:08:02,333
Just remember about that
per unit trick or tip I guess

148
00:08:03,033 --> 00:08:07,433
because if you forget about that you can, 
you know, make the wrong conclusions.

149
00:08:08,200 --> 00:08:10,666
Otherwise it's all pretty simple.

150
00:08:10,666 --> 00:08:13,666
The last thing
that I wanted to mention here is that

151
00:08:13,833 --> 00:08:16,833
you can see that every time
we run a model, the coefficients change.

152
00:08:17,200 --> 00:08:21,500
So what that is telling us
is that coefficients actually only talk

153
00:08:21,500 --> 00:08:24,566
about the additional effect
of every single variable,

154
00:08:25,166 --> 00:08:27,266
given that the other variables
are already in place.

155
00:08:27,266 --> 00:08:31,166
So for instance, in this example,
your coefficient of marketing,

156
00:08:31,166 --> 00:08:33,266
which is 0.029

157
00:08:33,266 --> 00:08:36,366
is means that given that R&D spend

158
00:08:36,366 --> 00:08:39,833
is already in your model and it's, fixed,

159
00:08:40,433 --> 00:08:45,000
then marketing spend adds additional,
you know, contributes

160
00:08:45,000 --> 00:08:49,700
this additional effect of 0.0299,
meaning that if you were to

161
00:08:50,533 --> 00:08:52,833
run a different model
and take out R&D, spend,

162
00:08:52,833 --> 00:08:54,733
then the coefficient
would be completely different.

163
00:08:54,733 --> 00:08:57,266
And that's what we see here,
that when we take our marketing,

164
00:08:57,266 --> 00:09:00,466
the coefficient for
and spend is, taken out.

165
00:09:00,466 --> 00:09:03,200
So that's another thing
you should remember about that.

166
00:09:03,200 --> 00:09:05,000
Coefficients merely

167
00:09:06,233 --> 00:09:07,066
portray the

168
00:09:07,066 --> 00:09:10,333
additional effect that every single
variable brings into the model.

169
00:09:10,900 --> 00:09:13,666
Hope you enjoyed this tutorial and
I look forward to seeing you next time.

170
00:09:13,666 --> 00:09:15,466
Until then, happy analyzing!