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![Page 1: 1 Re-expressing Data Chapter 6 – Normal Model –What if data do not follow a Normal model? Chapters 8 & 9 – Linear Model –What if a relationship between.](https://reader035.fdocuments.in/reader035/viewer/2022081516/56649d745503460f94a5501b/html5/thumbnails/1.jpg)
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Re-expressing Data Chapter 6 – Normal Model
–What if data do not follow a Normal model?
Chapters 8 & 9 – Linear Model–What if a relationship between
two variables is not linear?
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Re-expressing Data Re-expression is another
name for changing the scale of (transforming) the data.
Usually we re-express the response variable, Y.
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Goals of Re-expression Goal 1 – Make the distribution
of the re-expressed data more symmetric.
Goal 2 – Make the spread of the re-expressed data more similar across groups.
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Goals of Re-expression Goal 3 – Make the form of a
scatter plot more linear. Goal 4 – Make the scatter in
the scatter plot more even across all values of the explanatory variable.
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Ladder of PowersPower: 2Re-expression:Comment: Use on left skewed
data.
2y
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Ladder of PowersPower: 1Re-expression:Comment: No re-expression.
Do not re-express the data if they are already well behaved.
y
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Ladder of PowersPower: ½ Re-expression:Comment: Use on count data
or when scatter in a scatter plot tends to increase as the explanatory variable increases.
y
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Ladder of PowersPower: “0” Re-expression: Comments: Not really the “0”
power. Use on right skewed data. Measurements cannot be negative or zero.
ylog
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Ladder of PowersPower: –½, –1 Re-expression: Comments: Use on right
skewed data. Measurements cannot be negative or zero. Use on ratios.
yy
1,
1
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Goal 1 - Symmetry Data are obtained on the time
between nerve pulses along a nerve fiber.
Time is rounded to the nearest half unit where a unit is of a second.
– 30.5 represents
th
501
sec 61050530 ..
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.01
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.99
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nt0 10 20 30 40 50 60 70
Time ( sec)th
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Time – Nerve Pulses Distribution is skewed right. Sample mean (12.305) is much
larger than the sample median (7.5).
Many potential outliers. Data not from a Normal model.
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Sqrt(Time)
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Log(Time)
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Summary Time – Highly skewed to the
right. Sqrt(Time) – Still skewed right. Log(Time) –Fairly symmetric
and mounded in the middle.– Could have come from a Normal
model.
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Goal 3 – Straighten Up What is the relationship
between the temperature of coffee and the time since it was poured?–Y, temperature ( oF)–X, time (minutes)
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mp
0 10 20 30 40 50 60
Time (min)
Bivariate Fit of Temp By Time (min)
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Cooling Coffee There is a general negative
association – as time since the coffee was poured increases the temperature of the coffee decreases.
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Linear Model
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Tem
p (F
)
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Time (min)
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Linear Model Fit Summary
– Predicted Temp = 176.7 – 1.56*Time
– On average, temperature decreases 1.56 oF per minute.
– R2 = 0.99, 99% of the variation in temperature is explained by the linear relationship with time.
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Plot of Residuals
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esid
ual
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Time (min)
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Curved Pattern There is a clear pattern in the
plot of residuals versus time.–Under predict, over predict,
under predict. The linear fit is very good,
but we can do better.
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4.5
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Log(
Tem
p)
-10 0 10 20 30 40 50 60
Time (min)
Linear Fit
Bivariate Fit of Log(Temp) By Time (min)
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Log(Temp) by Time Summary
– Predicted Log(Temp) = 5.1946 –0.0114*Time
–On average, log temperature decreases 0.0114 log(oF) per minute.
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Plot of Residuals
-0.010
-0.005
0.000
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Res
idua
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Time (min)
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Interpretation There is a random scatter of
points around the zero line. The linear model relating
Log(Temp) to Time is the best we can do.
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Original Scale? Predicted Log(Temp) = 5.1946 –
0.0114*Time Predicted Temp =
180.3*e–0.0114*Time
– Predicted temp at time=0, 180.3 oF– The predicted temp in one more minute
is the predicted temp now multiplied by e–0.0114 = 0.98866
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JMP Method 1
–Create a new column in JMP, Log(Temp): Cols – Formula –Transcendental – Log.
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JMP Method 1 (continued)
–Fit Y by XY – Log(Temp)X – Time
–Fit Linear
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JMP Method 2
–Fit Y by XY – TempX – Time
–Fit SpecialTransform Y – Log