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Multiple Linear Regression

- Dingcai Cao
- University of Chicago
- Feb 9, 2006

On Tuesday Simple Linear Regression

Linear regression

- Introduction
- Linear regression is a general method for

estimating/describing association between a

continuous outcome variable (dependent) and one

or multiple predictors in one equation. - Statistical model Least squares Estimates
- Diagnosis
- Normality
- Constant variance
- Outliers

Today The relationship b/w simple linear

regression and ANOVA Multiple linear regression

Linear regression

Today The relationship b/w simple linear

regression and ANOVA Multiple linear regression

Linear Regression vs. ANOVA

ANOVA Dependent Continuous Independent

Categorical

Linear regression Dependent Continuous

Independent Continuous/ Categorical

Linear models

ANOVA and regression are the same thing!!!

Linear Regression vs. ANOVA

Scientific Question Is there any difference in

the loneliness between female and male H0

Female Male H1 Female Male

Student t test or ANOVA

Linear Regression vs. ANOVA

ANOVA WITH SOLUTION PROC GLM DATA

MYLIB.LONELINESS CLASS GENDER MODEL LONELY3

GENDER/SOLUTION RUN LINEAR REGRESSION USING

GLM PROC GLM DATA MYLIB.LONELINESS MODEL

LONELY3 GENDER RUN

Linear Regression vs. ANOVA

ANOVA

REGRESSION

Dependent Variable loneliness

Sum of Source

DF Squares Mean Square F

Value Pr gt F Model

1 4.902852

4.902852 2.24 0.1347

Error 498

1087.709443 2.184156

Corrected Total

499 1092.612294

R-Square Coeff Var Root MSE lonely3

Mean

0.004487 29.18049 1.477889

5.064648 Source

DF Type I SS Mean Square

F Value Pr gt F

gender 1 4.90285179

4.90285179 2.24 0.1347

Source DF Type

III SS Mean Square F Value Pr gt F

gender 1

4.90285179 4.90285179 2.24

0.1347

Standard

Parameter Estimate

Error t Value Pr gt t

Intercept 4.931462122

B 0.11077245 44.52 lt.0001

gender 1

0.206809959 B 0.13803488 1.50

0.1347 gender 2

0.000000000 B . .

.

Sum of

Source DF Squares

Mean Square F Value Pr gt F

Model 1

4.902852 4.902852 2.24 0.1347

Error

498 1087.709443 2.184156

Corrected Total

499 1092.612294

R-Square Coeff Var Root MSE

lonely3 Mean

0.004487 29.18049 1.477889

5.064648 Source

DF Type I SS Mean Square

F Value Pr gt F

gender 1 4.90285179

4.90285179 2.24 0.1347

Source DF Type

III SS Mean Square F Value Pr gt F

gender 1

4.90285179 4.90285179 2.24

0.1347

Standard

Parameter Estimate

Error t Value Pr gt t

Intercept 5.345082039

0.19850165 26.93 lt.0001

gender -0.206809959

0.13803488 -1.50 0.1347

Linear Regression vs. ANOVA

In the example, we show that ANOVA is a special

case of linear regression.

What if there are more than 2 groups in the

ANOVA

Linear Regression vs. ANOVA

Dummy variable for categorical data Outcome Y

(continuous) Predictor X (categorical)

X Group 1 Group 2 Group 3

Linear Regression

- Linear regression is a general method for

estimating/describing association between a

continuous outcome variable (dependent) and one

or multiple predictors in one equation. - One predictor Simple linear regression
- Multiple predictors Multiple linear regression

Statistical Model

Simple linear regression

Multiple linear regression

x

Example The academic performance of schools

Variable

Type Len Pos Label

---------------------------------------------

--------------------------------

11 acs_46 Num 3 39

avg class size 4-6

10 acs_k3 Num 3

36 avg class size k-3

3 api00 Num

4 12 api 2000

4 api99 Num

4 16 api 1999

17 avg_ed

Num 8 57 avg parent ed

15

col_grad Num 3 51 parent college

grad

22 collcat Num 8 78

2 dnum Num 4 8

district number

7 ell Num 3

27 english language learners

19 emer Num

3 68 pct emer credential

20 enroll Num

4 71 number of students

18 full

Num 3 65 pct full credential

16

grad_sch Num 3 54 parent grad

school

5 growth Num 4 20 growth

1999 to 2000

13 hsg Num 3 45

parent hsg

21 mealcat Num 3

75 Percentage free meals in 3 categories

6 meals Num

3 24 pct free meals

9 mobility Num

3 33 pct 1st year in school

12 not_hsg

Num 3 42 parent not hsg

1 snum

Num 8 0 school number

14

some_col Num 3 48 parent some

college

8 yr_rnd Num 3 30 year

round school

PROC CONTENTS OUTPUT BY SAS

Example The academic performance of schools

PROC CORR DATA MYLIB.ELEMAPI2 VAR API00 ELL

MEALS YR_RND MOBILITY ACS_K3 ACS_46 FULL EMER

ENROLL RUN

PROC CORR OUTPUT BY SAS

Pearson Correlation Coefficients

Prob gt r

under H0 Rho0

Number of Observations

api00 ell

meals yr_rnd mobility acs_k3 acs_46

full emer enroll api00

1.00000 -0.76763 -0.90070 -0.47544

-0.20641 0.17100 0.23291 0.57441 -0.58273

-0.31817 api 2000

lt.0001 lt.0001 lt.0001 lt.0001 0.0006

lt.0001 lt.0001 lt.0001 lt.0001 ell

-0.76763 1.00000 0.77238

0.49793 -0.02046 -0.05565 -0.17330 -0.48476

0.47218 0.40302 english language learners

lt.0001 lt.0001 lt.0001 0.6837

0.2680 0.0005 lt.0001 lt.0001 lt.0001

meals -0.90070 0.77238

1.00000 0.41845 0.21665 -0.18797 -0.21309

-0.52756 0.53304 0.24103 pct free meals

lt.0001 lt.0001 lt.0001

lt.0001 0.0002 lt.0001 lt.0001 lt.0001

lt.0001 yr_rnd -0.47544

0.49793 0.41845 1.00000 0.03479 0.02270

-0.04207 -0.39771 0.43472 0.59182 year

round school lt.0001 lt.0001

lt.0001 0.4883 0.6517 0.4032

lt.0001 lt.0001 lt.0001 mobility

-0.20641 -0.02046 0.21665 0.03479

1.00000 0.04014 0.12769 0.02521 0.05961

0.10502 pct 1st year in school lt.0001

0.6837 lt.0001 0.4883 0.4245

0.0110 0.6156 0.2348 0.0360 acs_k3

0.17100 -0.05565 -0.18797

0.02270 0.04014 1.00000 0.27078 0.16057

-0.11033 0.10890 avg class size k-3

0.0006 0.2680 0.0002 0.6517 0.4245

lt.0001 0.0013 0.0277 0.0298

acs_46 0.23291 -0.17330

-0.21309 -0.04207 0.12769 0.27078 1.00000

0.11773 -0.12446 0.02829 avg class size

4-6 lt.0001 0.0005 lt.0001

0.4032 0.0110 lt.0001 0.0190

0.0131 0.5741 full

0.57441 -0.48476 -0.52756 -0.39771 0.02521

0.16057 0.11773 1.00000 -0.90568 -0.33769

pct full credential lt.0001 lt.0001

lt.0001 lt.0001 0.6156 0.0013 0.0190

lt.0001 lt.0001 emer

-0.58273 0.47218 0.53304 0.43472

0.05961 -0.11033 -0.12446 -0.90568 1.00000

0.34309 pct emer credential lt.0001

lt.0001 lt.0001 lt.0001 0.2348 0.0277

0.0131 lt.0001 lt.0001 enroll

-0.31817 0.40302 0.24103

0.59182 0.10502 0.10890 0.02829 -0.33769

0.34309 1.00000 number of students

lt.0001 lt.0001 lt.0001 lt.0001 0.0360

0.0298 0.5741 lt.0001 lt.0001

Example The academic performance of schools

Dependent Variable api00 api 2000

Analysis of Variance

Sum of

Mean Source

DF Squares Square F

Value Pr gt F Model

9 6740702

748967 232.41 lt.0001

Error 385 1240708

3222.61761

Corrected Total 394

7981410

Root MSE

56.76810 R-Square 0.8446

Dependent Mean

648.65063 Adj R-Sq 0.8409

Coeff Var

8.75172

Parameter

Estimates

Parameter

Standard Variable Label

DF Estimate

Error t Value Pr gt t

Intercept Intercept 1

758.94179 62.28601 12.18 lt.0001

ell english language

learners 1 -0.86007 0.21063

-4.08 lt.0001 meals

pct free meals 1 -2.94822

0.17035 -17.31 lt.0001

yr_rnd year round school 1

-19.88875 9.25844 -2.15

0.0323 mobility pct 1st year

in school 1 -1.30135 0.43621

-2.98 0.0030 acs_k3

avg class size k-3 1 1.31870

2.25268 0.59 0.5586

acs_46 avg class size 4-6 1

2.03246 0.79832 2.55

0.0113 full pct full

credential 1 0.60972

0.47582 1.28 0.2008

emer pct emer credential 1

-0.70662 0.60541 -1.17 0.2439

enroll number of students

1 -0.01216 0.01679

-0.72 0.4693

Diagnosis Normal Distribution

Histogram

Boxplot

Normal Probability Plot 150

2 0

150

.

8

. 7

.

15

. 29

.

33

. 40

-----

.

57

-10

60 -----

-10

. 49

-----

.

39

.

26

.

17

.

7

. 3

.

2

-170

1 0 -170

------------------------

----------------

------------------------ may

represent up to 2 counts

-2 -1 0

1 2

Diagnosis Constant Variance

Residual vs fitted value plot

Diagnosis Outliers

Model Selection

Dependent Variable api00 api 2000

Analysis of Variance

Sum of

Mean Source

DF Squares Square F

Value Pr gt F Model

9 6740702

748967 232.41 lt.0001

Error 385 1240708

3222.61761

Corrected Total 394

7981410

Root MSE

56.76810 R-Square 0.8446

Dependent Mean

648.65063 Adj R-Sq 0.8409

Coeff Var

8.75172

Parameter

Estimates

Parameter

Standard Variable Label

DF Estimate

Error t Value Pr gt t

Intercept Intercept 1

758.94179 62.28601 12.18 lt.0001

ell english language

learners 1 -0.86007 0.21063

-4.08 lt.0001 meals

pct free meals 1 -2.94822

0.17035 -17.31 lt.0001

yr_rnd year round school 1

-19.88875 9.25844 -2.15

0.0323 mobility pct 1st year

in school 1 -1.30135 0.43621

-2.98 0.0030 acs_k3

avg class size k-3 1 1.31870

2.25268 0.59 0.5586

acs_46 avg class size 4-6 1

2.03246 0.79832 2.55

0.0113 full pct full

credential 1 0.60972

0.47582 1.28 0.2008

emer pct emer credential 1

-0.70662 0.60541 -1.17 0.2439

enroll number of students

1 -0.01216 0.01679

-0.72 0.4693

Model Selection

Forward Model Selection Starting with the null

model, add variables sequentially. Backward

Model Selection Starting with the full model,

delete variables with large P-values

sequentially. Stepwise Model Selection Combinati

on of Backward/Forward methods.

PROC REG DATA MYLIB.ELEMAPI2 M1 MODEL API00

ELL MEALS YR_RND MOBILITY ACS_K3 ACS_46 FULL

EMER ENROLL/SELECTION FORWARD M2 MODEL API00

ELL MEALS YR_RND MOBILITY ACS_K3 ACS_46 FULL

EMER ENROLL/SELECTION BACKWARD M3 MODEL

API00 ELL MEALS YR_RND MOBILITY ACS_K3 ACS_46

FULL EMER ENROLL/SELECTION STEPWISE RUN

Forward Model Selection

Forward Selection Step 1

Variable meals

Entered R-Square 0.8104 and C(p) 78.6700

Analysis of Variance

Sum of

Mean Source

DF Squares Square

F Value Pr gt F

Model 1 6467843

6467843 1679.39 lt.0001

Error 393 1513567

3851.31515

Corrected Total 394

7981410

Parameter Standard

Variable Estimate Error Type

II SS F Value Pr gt F

Intercept 890.33789 6.67326

68555604 17800.6 lt.0001

meals -4.01693 0.09802

6467843 1679.39 lt.0001

Forward Model Selection

Forward Selection Step 2

Variable emer Entered R-Square

0.8256 and C(p) 42.9320

Analysis of

Variance

Sum of Mean

Source

DF Squares Square F Value Pr

gt F Model

2 6589458 3294729

927.86 lt.0001 Error

392 1391952

3550.89695

Corrected Total 394

7981410

Parameter Standard

Variable Estimate Error Type

II SS F Value Pr gt F

Intercept 891.69625 6.41191

68674840 19340.1 lt.0001

meals -3.66331 0.11185

3809235 1072.75 lt.0001

emer -1.78671 0.30530

121615 34.25 lt.0001

Forward Model Selection

Forward Selection Step 3

Variable ell Entered R-Square

0.8352 and C(p) 21.2736

Analysis of

Variance

Sum of Mean

Source

DF Squares Square F Value Pr

gt F Model

3 6665700 2221900

660.30 lt.0001 Error

391 1315710

3364.98682

Corrected Total 394

7981410

Parameter Standard

Variable Estimate Error Type

II SS F Value Pr gt F

Intercept 887.25446 6.31117

66505925 19764.1 lt.0001

ell -0.88928 0.18683

76242 22.66 lt.0001

meals -3.16584 0.15092

1480643 440.01 lt.0001

emer -1.61167 0.29947

97463 28.96 lt.0001

Forward Model Selection

Forward Selection Step 4

Variable yr_rnd Entered

R-Square 0.8380 and C(p) 16.1280

Analysis of Variance

Sum of

Mean Source

DF Squares Square F

Value Pr gt F Model

4 6688728

1672182 504.49 lt.0001

Error 390 1292682

3314.56983

Corrected Total 394

7981410

Parameter Standard

Variable Estimate Error Type

II SS F Value Pr gt F

Intercept 885.52859 6.29784

65531336 19770.7 lt.0001

ell -0.73721 0.19419

47770 14.41 0.0002

meals -3.17476 0.14983

1488241 449.00 lt.0001

yr_rnd -21.74359 8.24936

23028 6.95 0.0087

emer -1.40732 0.30716

69579 20.99 lt.0001

Forward Model Selection

Forward Selection Step 5

Variable mobility Entered

R-Square 0.8405 and C(p) 11.9797

Analysis of Variance

Sum of

Mean Source

DF Squares Square F

Value Pr gt F Model

5 6708541

1341708 410.04 lt.0001

Error 389 1272868

3272.15552

Corrected Total 394

7981410

Parameter Standard

Variable Estimate Error Type

II SS F Value Pr gt F

Intercept 900.34592 8.68410

35172503 10749.0 lt.0001

ell -0.88118 0.20162

62504 19.10 lt.0001

meals -3.03351 0.15955

1182889 361.50 lt.0001

yr_rnd -20.98176 8.20226

21412 6.54 0.0109

mobility -1.01603 0.41290

19814 6.06 0.0143

emer -1.44072 0.30549

72777 22.24 lt.0001

Forward Model Selection

Forward Selection Step 6

Variable acs_46 Entered

R-Square 0.8434 and C(p) 6.9282

Analysis of Variance

Sum of

Mean Source

DF Squares Square F

Value Pr gt F Model

6 6731265

1121878 348.19 lt.0001

Error 388 1250144

3222.02150

Corrected Total 394

7981410

Parameter Standard

Variable Estimate Error Type

II SS F Value Pr gt F

Intercept 838.88765 24.69433

3718275 1154.02 lt.0001

ell -0.89299 0.20012

64158 19.91 lt.0001

meals -2.95860 0.16081

1090573 338.47 lt.0001

yr_rnd -22.35084 8.15549

24200 7.51 0.0064

mobility -1.22139 0.41695

27648 8.58 0.0036

acs_46 2.06035 0.77582

22724 7.05 0.0082

emer -1.42212 0.30322

70872 22.00 lt.0001

Forward Model Selection

Forward Selection Step 7

Variable full Entered R-Square

0.8442 and C(p) 6.8016

Analysis of

Variance

Sum of Mean

Source

DF Squares Square F Value Pr

gt F Model

7 6738119 962588

299.63 lt.0001 Error

387 1243291

3212.63859

Corrected Total 394

7981410

Parameter Standard

Variable Estimate Error Type

II SS F Value Pr gt F

Intercept 770.54568 52.89162

681844 212.24 lt.0001

ell -0.88545 0.19989

63036 19.62 lt.0001

meals -2.93428 0.16144

1061304 330.35 lt.0001

yr_rnd -22.76066 8.14844

25066 7.80 0.0055

mobility -1.35103 0.42570

32358 10.07 0.0016

acs_46 2.11781 0.77569

23948 7.45 0.0066

full 0.68305 0.46767

6853.20941 2.13 0.1450

emer -0.66301 0.60150

3903.28850 1.21 0.2710

Forward Model Selection

Forward Selection Step 8

Variable enroll Entered R-Square

0.8444 and C(p) 8.3427

Analysis of

Variance

Sum of Mean

Source

DF Squares Square F Value Pr

gt F Model

8 6739598 842450

261.86 lt.0001 Error

386 1241812

3217.12983

Corrected Total 394

7981410

Parameter Standard

Variable Estimate Error Type

II SS F Value Pr gt F

Intercept 777.22985 53.83881

670466 208.40 lt.0001

ell -0.84478 0.20883

52648 16.36 lt.0001

meals -2.96498 0.16778

1004713 312.30 lt.0001

yr_rnd -19.80050 9.24933

14743 4.58 0.0329

mobility -1.29000 0.43540

28240 8.78 0.0032

acs_46 2.13851 0.77683

24380 7.58 0.0062

full 0.64876 0.47072

6110.87487 1.90 0.1689

emer -0.67253 0.60209

4013.89996 1.25 0.2647

enroll -0.01134 0.01672

1479.01726 0.46 0.4982

Forward Model Selection

No other variable met

the 0.5000 significance level for entry into the

model.

Summary of Forward Selection

Variable

Number Partial Model Step

Entered Label Vars In

R-Square R-Square C(p) F Value

Pr gt F 1 meals pct free

meals 1 0.8104

0.8104 78.6700 1679.39 lt.0001

2 emer pct emer credential

2 0.0152 0.8256 42.9320

34.25 lt.0001 3 ell

english language learners 3 0.0096

0.8352 21.2736 22.66 lt.0001

4 yr_rnd year round school

4 0.0029 0.8380 16.1280

6.95 0.0087 5 mobility pct

1st year in school 5 0.0025

0.8405 11.9797 6.06 0.0143

6 acs_46 avg class size 4-6

6 0.0028 0.8434 6.9282

7.05 0.0082 7 full pct

full credential 7 0.0009

0.8442 6.8016 2.13 0.1450

8 enroll number of students

8 0.0002 0.8444 8.3427

0.46 0.4982

Collinearity/Mulilinearity

- Refer to the dependency in the predictors. That

is, a predictor is nearly a linear combination of

other predictors in the model. - Why do we care
- Difficult to draw conclusion based on the results
- YX1X2, X1 2X2,
- Unstable estimates and large standard errors
- It is always important to look at the correlation

matrix before model fitting.

Collinearity/Mulilinearity

Statistically, collinearity is measured by two

values Variance inflation factor

(VIF) Tolerance (TOL) TOL(X1) 1 R2(X1X2,

X3, , Xk) TOL(X2) 1 R2(X2X1, X3, ,

Xk) TOL(Xi) 1 R2(Xi All Xj, j i) VIF

1/TOL The lowest value of VIF is 1.0 VIF gt 10,

Strong indication of collinearity

Collinearity SAS Option

PROC REG DATA MYLIB.ELEMAPI2 MODEL API00

ELL MEALS YR_RND MOBILITY ACS_K3 ACS_46 FULL EMER

ENROLL/VIF TOL OUTPUT OUT C P PRED L95

L95 U95 U95 R RESID COOKD COOKD RUN

Parameter Estimates

Parameter

Standard

Variance Variable Label

DF Estimate Error t Value

Pr gt t Tolerance Inflation

Intercept Intercept 1

758.94179 62.28601 12.18 lt.0001

. 0 ell

english language learners 1 -0.86007

0.21063 -4.08 lt.0001 0.30090

3.32338 meals pct free meals

1 -2.94822 0.17035

-17.31 lt.0001 0.27706 3.60929

yr_rnd year round school 1

-19.88875 9.25844 -2.15 0.0323

0.53272 1.87716 mobility pct

1st year in school 1 -1.30135

0.43621 -2.98 0.0030 0.76279

1.31098 acs_k3 avg class size k-3

1 1.31870 2.25268 0.59

0.5586 0.85399 1.17098 acs_46

avg class size 4-6 1

2.03246 0.79832 2.55 0.0113

0.86593 1.15483 full pct full

credential 1 0.60972

0.47582 1.28 0.2008 0.16289

6.13907 emer pct emer credential

1 -0.70662 0.60541 -1.17

0.2439 0.16344 6.11858 enroll

number of students 1

-0.01216 0.01679 -0.72 0.4693

0.56555 1.76820

Goodness of Fit

R2 The larger, the better Adjusted R2 The

larger, the better Root MSE The smaller, the

better Most frequently used statistic (probably)

is Cp p 1 number of independent variables

(k)

Model Comparison

Full Model 1RSS1 (df1) Simple Model 2RSS2

(df2) F(df2 df1, df1) (RSS2

RSS1)/(df2-df1)/(RSS1/df1))

Model Comparison

Analysis of Variance

Sum

of Mean

Source DF Squares

Square F Value Pr gt F

Model 8 6739598

842450 261.86 lt.0001

Error 386

1241812 3217.12983

Corrected Total 394

7981410

Parameter Standard

Variable Estimate Error Type

II SS F Value Pr gt F

Intercept 777.22985 53.83881

670466 208.40 lt.0001

ell -0.84478 0.20883

52648 16.36 lt.0001

meals -2.96498 0.16778

1004713 312.30 lt.0001

yr_rnd -19.80050 9.24933

14743 4.58 0.0329

mobility -1.29000 0.43540

28240 8.78 0.0032

acs_46 2.13851 0.77683

24380 7.58 0.0062

full 0.64876 0.47072

6110.87487 1.90 0.1689

emer -0.67253 0.60209

4013.89996 1.25 0.2647

enroll -0.01134 0.01672

1479.01726 0.46 0.4982

Model Comparison

Analysis of Variance

Sum

of Mean

Source DF Squares

Square F Value Pr gt F

Model 5 6708541

1341708 410.04 lt.0001

Error 389

1272868 3272.15552

Corrected Total 394

7981410

Parameter Standard

Variable Estimate Error Type

II SS F Value Pr gt F

Intercept 900.34592 8.68410

35172503 10749.0 lt.0001

ell -0.88118 0.20162

62504 19.10 lt.0001

meals -3.03351 0.15955

1182889 361.50 lt.0001

yr_rnd -20.98176 8.20226

21412 6.54 0.0109

mobility -1.01603 0.41290

19814 6.06 0.0143

emer -1.44072 0.30549

72777 22.24 lt.0001

Model Comparison

Full Model 1 api00 ell meals yr_rnd

mobility acs_46 full emer enroll RSS1

1241812 df1 386 Simple Model 2RSS2

(df2) api00 ell meals yr_rnd mobility

emer RSS2 1272868 df2 389 F(df2 df1, df1)

(RSS2 RSS1)/(df2-df1)/(RSS1/df1)) F(3,

386) (1272868 -1241812 )/3/ (1241812

/386) 3.22 P 0.00022

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presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

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