Its a class presentation. its contain enough knowledge.

1. Muhammad Qasim
Aroj Bashir
Dept. Environmental Sciences
University of Gujrat

2. What is Regression
What is Multi-Regression
What is ANOVA & Two Way ANOVA
Difference Between Multi-Regression & Two way
ANOVA.
Example of Multi Regression
Example of Two ay ANOVA

3. Regression is the relationship between
one dependent variable (usually denoted
by Y) and a series of other changing
variables (known as independent
variables).
two purposes of regression procedures—
prediction and explanation.
Example: The price of a commodity, interest
rates, particular industries or sectors
influence the price movement of an asset.
Types of Regression:
1- Liner Regression:
Linear regression uses one independent
variable to explain and/or predict the
outcome of Y. General Form is,
Linear Regression: Y = a + bX + u

4. Multiple regression uses two or more
independent variables to predict the
outcome.
Y=a + b1X1 + b2X2 + b3X3
There are several different kinds of
multiple regressions—simultaneous,
stepwise, and hierarchical multiple
regression.
In simultaneous (aka, standard) multiple
regression, all the independent variables
are considered at the same time.

5. Stepwise multiple regression
In the computer determines the order
in which the independent variables
become part of the equation.
In hierarchical multiple regression
The researcher determines the order
that the independent variables are
entered in the equation. The order for
entering the variables should be based
on theory.
It is possible to use nominal level data
as an independent variable. For
example, gender (female and male)
can be used as an independent
variable.

6. Analysis of variance is a method for
splitting the total variation of our data
into meaningful components that
measure different source of variation.
Two meaningful components
1- Measuring the variation due to
experimental error
2- Measuring the variation due to
experimental error plus any variation
due to the different variable.
Two Types:
OneWay ANOVA---Single Criterion
TwoWay ANOVA--- Two Criterion

7. Two-way (or multi-way) ANOVA is an
appropriate analysis method for a study with
a quantitative outcome and two (or more)
categorical explanatory variables.
Suppose we now have two categorical
explanatory variables
Is there a significant X1 effect?
Is there a significant X2 effect?
Are there significant interaction effects?
If X1 has k levels and X2 has m levels, then
the analysis is often referred to as a “k by m
ANOVA” or “k x mANOVA”

8. S.No Multiple Regression TwoWay ANOVA
1 One or more continuous
predictor variables
One or more categorical
variables
2 Shows relationship
between variables
Compares means
3 Regression technique
produces adjusted F or t
statistics
ANOVA technique produces
adjusted F statistics, and depending
on the software
4 Predict the duration of child
feeding in weeks using
mother's age as a predictor
variable
Duration of child feeding in weeks
using mother's marital status (single,
married, divorced, widowed),

9. Example:
In an experiment conducted to
determine which of 3
missile system is preferable,
the propellant burning rate
for 24 static firings was
measured. Four propellant
type were used. The
experiment yielded
duplicate observations of
burnings rates at each
combinations of the
treatments. The data, after
coding, were recorded as
follow?
Missile
System
Propellant Type
b1 b2 b3 b4
a1
34 30.1 29.8 29
32.7 32.8 26.7 28.9
a2
32 30.2 28.7 27.6
33.2 29.8 28.1 27.8
a3
28.4 27.3 29.7 28.8
29.3 28.9 27.3 29.1

10. Solution
1. Hypothesis
2. Level of Significance
3. Critical Region
4. Test Statics
5. Computations
6. Decision

11. Hypothesis
’: There is no difference in the mean propellant burning rate when
different missile systems are used.
A- H0
’’: There is no difference in the mean propellant burning rate of the 4th
propellant types.
B -H0
’’’: There is no interaction between the different missile systems and the
different propellant types
Level of Significance
C- H0
0.05 is the level of significance
Critical Region
Value compression of computed f & tabulated f
f > 3.13, f > 3.52, f > 2.59

12. Test Statics
Sum of Square colum : SSC= Ƹci2/ni - (Ƹci)2/N
Sum of Square of Row : SSR= Ƹri2/mi - (Ƹci)2/N
Interaction : SS(RC)= Ƹ(viti)/r - (SSC+SSR+(Ƹci)2/N
Sum of Square Error: SSE= SST-SSC-SSR-SS(RC)
Sum of Square of Total : SST= Ƹxi2 - (Ƹci)2)/N

13. Computations
SSC= Ƹci2/ni - (Ƹci)2/N
= 126336.5/6 – 504384.04/24
=21056.08-21016.00167
= 40.0783333
Missile
System
Propellant Type
b1 b2 b3 b4 Ƹri Ƹri2
a1
34 30.1 29.8 29
244 59536
32.7 32.8 26.7 28.9
a2
32 30.2 28.7 27.6
237.4 56358.76
33.2 29.8 28.1 27.8
a3
28.4 27.3 29.7 28.8
228.8 52349.44
29.3 28.9 27.3 29.1
Ƹci 189.6 179.1 170.3 171.2 710.2168244.2
Ƹci2 35948.16 32076.81 29002.09 29309.44 126336.5

14. Computations
SSR= Ƹri2/mi - (Ƹci)2/N
= 168244.2/8 – 504384.04/24
=21030.525-21016.00167
= 14.52333
Missile
System
Propellant Type
b1 b2 b3 b4 Ƹri Ƹri2
a1
34 30.1 29.8 29
244 59536
32.7 32.8 26.7 28.9
a2
32 30.2 28.7 27.6
237.4 56358.
76
33.2 29.8 28.1 27.8
a3
28.4 27.3 29.7 28.8
228.8 52349.
44
29.3 28.9 27.3 29.1
Ƹci 189.6 179.1 170.3 171.2 710.2
168244
.2
Ƹci2 35948.
16
32076.
81
29002.
09
29309.
44
126336
.5

15. Computations
SS(RC)= Ƹ(viti)/r - (SSC+SSR+(Ƹci)2/N)
= 42185.54/ –(40.0783333 + 14.52333 +504384.04/2)
=21092.77-21070.6
= 22.1666667 Formulated Table
Missile
System
Propellant Type
A B C D
1 66.7 62.9 56.5 57.9
2 65.2 60 56.8 55.4
3 57.7 56.2 57 57.9
Ƹ(viti)/r : 42185.54

16. Computations
SST= Ƹxi2 - (Ƹci)2)/N
= (34.02+32.72+30.12+………28.82+29.12) -
504384.04/24
=21107.68 - 21016.00167
= 91.67833
Missile
System
Propellant
Type
b1 b2 b3 b4 Ƹri
Ƹri
2
a1
34 30.1 29.8 29
244 59536
32.7 32.8 26.7 28.9
a2
32 30.2 28.7 27.6 237.
4
56358.
76
33.2 29.8 28.1 27.8
a3
28.4 27.3 29.7 28.8 228.
8
52349.
44
29.3 28.9 27.3 29.1
Ƹci
189.
6
179.
1
170.
3
171.
2
710
.2
16824
4.2
Ƹci2 35948.
16
32076.
81
29002.
09
29309.
44
12633
6.5

17. Computation
SSE= SST-SSC-SSR-SS(RC)
= 91.67833- 40.0783333- 14.52333- 22.1666667
= 14.91
ANOVA TABLE
Source of
Variation
d.f
Sum of Square
(SS)
Mean
Sequare
(MS)
Computed
f
f
Crit.
SSC 3 40.0783333 13.35 17.024 3.13
SSR 2 14.52333 7.261 9.254 3.52
SS(RS) 6 22.1666667 3.694 4.7024 2.63
SSE 19 14.91 0.7847
SST 24 91.67833 3.81993

18. ANOVA TABLE
Source of
Variation
d.f
Sum of Square
(SS)
Mean
Sequare
(MS)
Computed
f
f
Crit.
SSC 3 40.0783333 13.35 17.024 3.13
SSR 2 14.52333 7.261 9.254 3.52
SS(RS) 6 22.1666667 3.694 4.7024 2.63
SSE 19 14.91 0.7847
SST 24 91.67833 3.81993
Decision
Our null hypothesis accepted and mean propellant rates are
used