C orrelation and regression II. Regression using transformations.
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C orrelation and regression II. Regression using transformations. Scatterplot Relationship between two contin uo us variables. Scatterplot Other examples. Example II. - PowerPoint PPT Presentation
Transcript of C orrelation and regression II. Regression using transformations.
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Correlation and regression II. Regression using transformations. *
Biostat 9.
Krisztina Boda
Biostat 9.*ScatterplotRelationship between two continuous variables
Biostat 9.
Munka1
StudentHours studiedGrade
Jane870
Joe1080
Sue1275
Pat1990
Bob2085
Tom2595
Munka1
Grade
Munka2
Munka3
Krisztina Boda
Biostat 9.*ScatterplotOther examples
Biostat 9.
Krisztina Boda
Biostat 9.*Example II.Imagine that 6 students are given a battery of tests by a vocational guidance counsellor with the results shown in the following table:
Variables measured on the same individuals are often related to each other.
Biostat 9.
Krisztina Boda
Biostat 9.*Possible relationshipspositive correlationnegative correlationno correlation
Biostat 9.
Diagram1
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Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
math
math score
theater
Munka2
math
math score
theater
Munka3
math
math score
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retailing
xy
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2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
y
Diagram2
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Pat5130525550
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Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
math
math score
theater
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math
math score
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math
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1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
y
Diagram3
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Nameretailtheatermathlanguage
Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
math
math score
theater
Munka2
math
math score
theater
Munka3
math
math score
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math
math score
language
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retailing
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10.8414709848
1.20.932039086
1.40.98544973
1.60.999573603
1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
y
Krisztina Boda
Biostat 9.*Describing linear relationship with number: the coefficient of correlation (r).Also called Pearson coefficient of correlationCorrelation is a numerical measure of the strength of a linear association. The formula for coefficient of correlation treats x and y identically. There is no distinction between explanatory and response variable. Let us denote the two samples by x1,x2,xn and y1,y2,yn , the coefficient of correlation can be computed according to the following formula
Biostat 9.
Krisztina Boda
Biostat 9.*Karl PearsonKarl Pearson (27 March 1857 27 April 1936) established the discipline of mathematical statistics. http://en.wikipedia.org/wiki/Karl_Pearson
Biostat 9.
Krisztina Boda
Biostat 9.*Properties of rCorrelations are between -1 and +1; the value of r is always between -1 and 1, either extreme indicates a perfect linear association. -1r 1.a) If r is near +1 or -1 we say that we have high correlation.
b) If r=1, we say that there is perfect positive correlation. If r= -1, then we say that there is a perfect negative correlation.
c) A correlation of zero indicates the absence of linear association. When there is no tendency for the points to lie in a straight line, we say that there is no correlation (r=0) or we have low correlation (r is near 0 ).
Biostat 9.
Diagram4
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Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
math
math score
theater
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math
math score
theater
Munka3
math
math score
language
math
math score
language
math
math score
retailing
xy
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00
0.20.1986693308
0.40.3894183423
0.60.5646424734
0.80.7173560909
10.8414709848
1.20.932039086
1.40.98544973
1.60.999573603
1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
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30.1411200081
y
Diagram5
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adatok
Nameretailtheatermathlanguage
Pat5130525550
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Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
math
math score
theater
Munka2
math
math score
theater
Munka3
math
math score
language
math
math score
language
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xy
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1.60.999573603
1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
y
Diagram6
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math
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retailing
adatok
Nameretailtheatermathlanguage
Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
math
math score
theater
Munka2
math
math score
theater
Munka3
math
math score
language
math
math score
language
math
math score
retailing
xy
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00
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1.20.932039086
1.40.98544973
1.60.999573603
1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
y
Krisztina Boda
Biostat 9.*Calculated values of rpositive correlation, r=0.9989negative correlation, r=-0.9993no correlation, r=-0.2157
Biostat 9.
Diagram1
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adatok
Nameretailtheatermathlanguage
Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
math
math score
theater
Munka2
math
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theater
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math
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language
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xy
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0.20.1986693308
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1.60.999573603
1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
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y
Diagram2
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adatok
Nameretailtheatermathlanguage
Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
math
math score
theater
Munka2
math
math score
theater
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math
math score
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math
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language
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xy
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1.20.932039086
1.40.98544973
1.60.999573603
1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
y
Diagram3
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adatok
Nameretailtheatermathlanguage
Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
math
math score
theater
Munka2
math
math score
theater
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math
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math
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language
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0.20.1986693308
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1.20.932039086
1.40.98544973
1.60.999573603
1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
y
Krisztina Boda
Biostat 9.*ScatterplotOther examplesr=0.873r=0.018
Biostat 9.
Krisztina Boda
Biostat 9.*Correlation and causationa correlation between two variables does not show that one causes the other.
Biostat 9.
Krisztina Boda
Biostat 9.*When is a correlation high?What is considered to be high correlation varies with the field of application. The statistician must decide when a sample value of r is far enough from zero, that is, when it is sufficiently far from zero to reflect the correlation in the population.
Biostat 9.
Krisztina Boda
Biostat 9.*Testing the significance of the coefficient of correlationThe statistician must decide when a sample value of r is far enough from zero to be significant, that is, when it is sufficiently far from zero to reflect the correlation in the population.
Biostat 9.
Krisztina Boda
Biostat 9.*Testing the significance of the coefficient of correlation
The statistician must decide when a sample value of r is far enough from zero to be significant, that is, when it is sufficiently far from zero to reflect the correlation in the population. This test can be carried out by expressing the t statistic in terms of r. H0: =0 (greek rho=0, correlation coefficient in population = 0)Ha: 0 (correlation coefficient in population 0)Assumption: the two samples are drawn from a bivariate normal distributionIf H0 is true, then the following t-statistic has n-2 degrees of freedom
Biostat 9.
Krisztina Boda
Biostat 9.*Bivariate normal distributions=0 =0.4Biostat 9.*
Biostat 9.
Krisztina Boda
Biostat 9.*Testing the significance of the coefficient of correlation
The following t-statistic has n-2 degrees of freedom
Decision using statistical table: If |t|>t,n-2, the difference is significant at level, we reject H0 and state that the population correlation coefficient is different from 0.If |t| 2.776, we reject H0 and claim that there is a significant linear correlation between the two variables at 5 % level.
Biostat 9.
Krisztina Boda
Biostat 9.*Example 2., cont.
Biostat 9.
Krisztina Boda
Biostat 9.*Example 3.The correlation coefficient between math skill and theater skill was found r= -0.2157. Is significantly different from 0?H0: the correlation coefficient in population = 0, =0.Ha: the correlation coefficient in population is different from 0.Let's compute the test statistic:
Degrees of freedom: df=6-2=4The critical value in the table is t0.05,4 = 2.776. Because |-0.4418|=0.4418 < 2.776, we do not reject H0 and claim that there is no a significant linear correlation between the two variables at 5 % level.
Biostat 9.
Krisztina Boda
Biostat 9.*Example 3., cont.
Biostat 9.
Krisztina Boda
Biostat 9.**Significance of the correlationOther examplesr=0.873, ptake log yModel: y=a*10bxTake the logarithm of both sides:lg y =lga+bxso lg y is linear in x
Biostat 9.
adatok
Nameretailtheatermathlanguage
Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
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adatok
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xylg y
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11.90.278753601
240.6020599913
38.10.9084850189
4161.2041199827
xylog x
10.10
420.6020599913
83.010.903089987
163.91.2041199827
xylog xlog y
1200.3010299957
2160.30102999571.2041199827
3540.47712125471.7323937598
41280.60205999132.1072099696
xy1/x
11.11
20.450.5
30.3330.3333333333
40.230.25
50.19990.2
0
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0
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0
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log y
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Diagram7
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adatok
Nameretailtheatermathlanguage
Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
math
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transzformcik
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20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
0
0.1986693308
0.3894183423
0.5646424734
0.7173560909
0.8414709848
0.932039086
0.98544973
0.999573603
0.9738476309
0.9092974268
0.8084964038
0.6754631806
0.5155013718
0.3349881502
y
xylg y
01.10.0413926852
11.90.278753601
240.6020599913
38.10.9084850189
4161.2041199827
xylog x
10.10
420.6020599913
83.010.903089987
163.91.2041199827
xylog xlog y
1200.3010299957
2160.30102999571.2041199827
3540.47712125471.7323937598
41280.60205999132.1072099696
xy1/x
11.11
20.450.5
30.3330.3333333333
40.230.25
50.19990.2
y
x
y
y
x
log y
y
x
y
y
log10 x
y
y
x
y
log y
log x
log y
y
x
y
y
1/x
y
Diagram8
0.0413926852
0.278753601
0.6020599913
0.9084850189
1.2041199827
y
x
log y
adatok
Nameretailtheatermathlanguage
Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
math
math score
theater
nemlineris
math
math score
theater
transzformcik
math
math score
language
math
math score
language
math
math score
retailing
xy
-3-0.1411200081
-2.8-0.3349881502
-2.6-0.5155013718
-2.4-0.6754631806
-2.2-0.8084964038
-2-0.9092974268
-1.8-0.9738476309
-1.6-0.999573603
-1.4-0.98544973
-1.2-0.932039086
-1-0.8414709848
-0.8-0.7173560909
-0.6-0.5646424734
-0.4-0.3894183423
-0.2-0.1986693308
00
0.20.1986693308
0.40.3894183423
0.60.5646424734
0.80.7173560909
10.8414709848
1.20.932039086
1.40.98544973
1.60.999573603
1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
0
0.1986693308
0.3894183423
0.5646424734
0.7173560909
0.8414709848
0.932039086
0.98544973
0.999573603
0.9738476309
0.9092974268
0.8084964038
0.6754631806
0.5155013718
0.3349881502
y
xylg y
01.10.0413926852
11.90.278753601
240.6020599913
38.10.9084850189
4161.2041199827
xylog x
10.10
420.6020599913
83.010.903089987
163.91.2041199827
xylog xlog y
1200.3010299957
2160.30102999571.2041199827
3540.47712125471.7323937598
41280.60205999132.1072099696
xy1/x
11.11
20.450.5
30.3330.3333333333
40.230.25
50.19990.2
y
x
y
y
x
log y
y
x
y
y
log10 x
y
y
x
y
log y
log x
log y
y
x
y
y
1/x
y
Krisztina Boda
Biostat 9.*Logarithm relationship ->take log xModel: y=a+lgx
so y is linear in lg x
Biostat 9.
adatok
Nameretailtheatermathlanguage
Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
0
0
0
0
0
0
0
math
math score
theater
nemlineris
0
0
0
0
0
0
math
math score
theater
transzformcik
0
0
0
0
0
0
0
math
math score
language
0
0
0
0
0
0
0
math
math score
language
0
0
0
0
0
0
math
math score
retailing
xy
-3-0.1411200081
-2.8-0.3349881502
-2.6-0.5155013718
-2.4-0.6754631806
-2.2-0.8084964038
-2-0.9092974268
-1.8-0.9738476309
-1.6-0.999573603
-1.4-0.98544973
-1.2-0.932039086
-1-0.8414709848
-0.8-0.7173560909
-0.6-0.5646424734
-0.4-0.3894183423
-0.2-0.1986693308
00
0.20.1986693308
0.40.3894183423
0.60.5646424734
0.80.7173560909
10.8414709848
1.20.932039086
1.40.98544973
1.60.999573603
1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
0
0.1986693308
0.3894183423
0.5646424734
0.7173560909
0.8414709848
0.932039086
0.98544973
0.999573603
0.9738476309
0.9092974268
0.8084964038
0.6754631806
0.5155013718
0.3349881502
y
xylg y
01.10.0413926852
11.90.278753601
240.6020599913
38.10.9084850189
4161.2041199827
xylog x
10.10
420.6020599913
83.010.903089987
163.91.2041199827
xylog xlog y
1200.3010299957
2160.30102999571.2041199827
3540.47712125471.7323937598
41280.60205999132.1072099696
xy1/x
11.11
20.450.5
30.3330.3333333333
40.230.25
50.19990.2
0
0
0
0
0
y
x
y
0
0
0
0
0
y
x
log y
0
0
0
0
y
x
y
0
0
0
0
y
log10 x
y
0
0
0
0
y
x
y
0
0
0
0
log y
log x
log y
0
0
0
0
0
y
x
y
0
0
0
0
0
y
1/x
y
Diagram9
0.1
2
3.01
3.9
y
x
y
adatok
Nameretailtheatermathlanguage
Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
math
math score
theater
nemlineris
math
math score
theater
transzformcik
math
math score
language
math
math score
language
math
math score
retailing
xy
-3-0.1411200081
-2.8-0.3349881502
-2.6-0.5155013718
-2.4-0.6754631806
-2.2-0.8084964038
-2-0.9092974268
-1.8-0.9738476309
-1.6-0.999573603
-1.4-0.98544973
-1.2-0.932039086
-1-0.8414709848
-0.8-0.7173560909
-0.6-0.5646424734
-0.4-0.3894183423
-0.2-0.1986693308
00
0.20.1986693308
0.40.3894183423
0.60.5646424734
0.80.7173560909
10.8414709848
1.20.932039086
1.40.98544973
1.60.999573603
1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
0
0.1986693308
0.3894183423
0.5646424734
0.7173560909
0.8414709848
0.932039086
0.98544973
0.999573603
0.9738476309
0.9092974268
0.8084964038
0.6754631806
0.5155013718
0.3349881502
y
xylg y
01.10.0413926852
11.90.278753601
240.6020599913
38.10.9084850189
4161.2041199827
xylog x
10.10
420.6020599913
83.010.903089987
163.91.2041199827
xylog xlog y
1200.3010299957
2160.30102999571.2041199827
3540.47712125471.7323937598
41280.60205999132.1072099696
xy1/x
11.11
20.450.5
30.3330.3333333333
40.230.25
50.19990.2
y
x
y
y
x
log y
y
x
y
y
log10 x
y
y
x
y
log y
log x
log y
y
x
y
y
1/x
y
Diagram10
0.1
2
3.01
3.9
y
log10 x
y
adatok
Nameretailtheatermathlanguage
Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
math
math score
theater
nemlineris
math
math score
theater
transzformcik
math
math score
language
math
math score
language
math
math score
retailing
xy
-3-0.1411200081
-2.8-0.3349881502
-2.6-0.5155013718
-2.4-0.6754631806
-2.2-0.8084964038
-2-0.9092974268
-1.8-0.9738476309
-1.6-0.999573603
-1.4-0.98544973
-1.2-0.932039086
-1-0.8414709848
-0.8-0.7173560909
-0.6-0.5646424734
-0.4-0.3894183423
-0.2-0.1986693308
00
0.20.1986693308
0.40.3894183423
0.60.5646424734
0.80.7173560909
10.8414709848
1.20.932039086
1.40.98544973
1.60.999573603
1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
0
0.1986693308
0.3894183423
0.5646424734
0.7173560909
0.8414709848
0.932039086
0.98544973
0.999573603
0.9738476309
0.9092974268
0.8084964038
0.6754631806
0.5155013718
0.3349881502
y
xylg y
01.10.0413926852
11.90.278753601
240.6020599913
38.10.9084850189
4161.2041199827
xylog x
10.10
420.6020599913
83.010.903089987
163.91.2041199827
xylog xlog y
1200.3010299957
2160.30102999571.2041199827
3540.47712125471.7323937598
41280.60205999132.1072099696
xy1/x
11.11
20.450.5
30.3330.3333333333
40.230.25
50.19990.2
y
x
y
y
x
log y
y
x
y
y
log10 x
y
y
x
y
log y
log x
log y
y
x
y
y
1/x
y
Krisztina Boda
Biostat 9.*Power relationship ->take log x and log yModel: y=axbTake the logarithm of both sides:lg y =lga+b lgxso lgy is linear in lg x
Biostat 9.
adatok
Nameretailtheatermathlanguage
Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
0
0
0
0
0
0
0
math
math score
theater
nemlineris
0
0
0
0
0
0
math
math score
theater
transzformcik
0
0
0
0
0
0
0
math
math score
language
0
0
0
0
0
0
0
math
math score
language
0
0
0
0
0
0
math
math score
retailing
xy
-3-0.1411200081
-2.8-0.3349881502
-2.6-0.5155013718
-2.4-0.6754631806
-2.2-0.8084964038
-2-0.9092974268
-1.8-0.9738476309
-1.6-0.999573603
-1.4-0.98544973
-1.2-0.932039086
-1-0.8414709848
-0.8-0.7173560909
-0.6-0.5646424734
-0.4-0.3894183423
-0.2-0.1986693308
00
0.20.1986693308
0.40.3894183423
0.60.5646424734
0.80.7173560909
10.8414709848
1.20.932039086
1.40.98544973
1.60.999573603
1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
0
0.1986693308
0.3894183423
0.5646424734
0.7173560909
0.8414709848
0.932039086
0.98544973
0.999573603
0.9738476309
0.9092974268
0.8084964038
0.6754631806
0.5155013718
0.3349881502
y
xylg y
01.10.0413926852
11.90.278753601
240.6020599913
38.10.9084850189
4161.2041199827
xylog x
10.10
420.6020599913
83.010.903089987
163.91.2041199827
xylog xlog y
1200.3010299957
2160.30102999571.2041199827
3540.47712125471.7323937598
41280.60205999132.1072099696
xy1/x
11.11
20.450.5
30.3330.3333333333
40.230.25
50.19990.2
0
0
0
0
0
y
x
y
0
0
0
0
0
y
x
log y
0
0
0
0
y
x
y
0
0
0
0
y
log10 x
y
y
x
y
0
0
0
0
log y
log x
log y
0
0
0
0
0
y
x
y
0
0
0
0
0
y
1/x
y
Diagram11
2
16
54
128
y
x
y
adatok
Nameretailtheatermathlanguage
Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
math
math score
theater
nemlineris
math
math score
theater
transzformcik
math
math score
language
math
math score
language
math
math score
retailing
xy
-3-0.1411200081
-2.8-0.3349881502
-2.6-0.5155013718
-2.4-0.6754631806
-2.2-0.8084964038
-2-0.9092974268
-1.8-0.9738476309
-1.6-0.999573603
-1.4-0.98544973
-1.2-0.932039086
-1-0.8414709848
-0.8-0.7173560909
-0.6-0.5646424734
-0.4-0.3894183423
-0.2-0.1986693308
00
0.20.1986693308
0.40.3894183423
0.60.5646424734
0.80.7173560909
10.8414709848
1.20.932039086
1.40.98544973
1.60.999573603
1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
0
0.1986693308
0.3894183423
0.5646424734
0.7173560909
0.8414709848
0.932039086
0.98544973
0.999573603
0.9738476309
0.9092974268
0.8084964038
0.6754631806
0.5155013718
0.3349881502
y
xylg y
01.10.0413926852
11.90.278753601
240.6020599913
38.10.9084850189
4161.2041199827
xylog x
10.10
420.6020599913
83.010.903089987
163.91.2041199827
xylog xlog y
1200.3010299957
2160.30102999571.2041199827
3540.47712125471.7323937598
41280.60205999132.1072099696
xy1/x
11.11
20.450.5
30.3330.3333333333
40.230.25
50.19990.2
y
x
y
y
x
log y
y
x
y
y
log10 x
y
y
x
y
log y
log x
log y
y
x
y
y
1/x
y
Diagram12
0.3010299957
1.2041199827
1.7323937598
2.1072099696
log y
log x
log y
adatok
Nameretailtheatermathlanguage
Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
math
math score
theater
nemlineris
math
math score
theater
transzformcik
math
math score
language
math
math score
language
math
math score
retailing
xy
-3-0.1411200081
-2.8-0.3349881502
-2.6-0.5155013718
-2.4-0.6754631806
-2.2-0.8084964038
-2-0.9092974268
-1.8-0.9738476309
-1.6-0.999573603
-1.4-0.98544973
-1.2-0.932039086
-1-0.8414709848
-0.8-0.7173560909
-0.6-0.5646424734
-0.4-0.3894183423
-0.2-0.1986693308
00
0.20.1986693308
0.40.3894183423
0.60.5646424734
0.80.7173560909
10.8414709848
1.20.932039086
1.40.98544973
1.60.999573603
1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
0
0.1986693308
0.3894183423
0.5646424734
0.7173560909
0.8414709848
0.932039086
0.98544973
0.999573603
0.9738476309
0.9092974268
0.8084964038
0.6754631806
0.5155013718
0.3349881502
y
xylg y
01.10.0413926852
11.90.278753601
240.6020599913
38.10.9084850189
4161.2041199827
xylog x
10.10
420.6020599913
83.010.903089987
163.91.2041199827
xylog xlog y
1200.3010299957
2160.30102999571.2041199827
3540.47712125471.7323937598
41280.60205999132.1072099696
xy1/x
11.11
20.450.5
30.3330.3333333333
40.230.25
50.19990.2
y
x
y
y
x
log y
y
x
y
y
log10 x
y
y
x
y
log y
log x
log y
y
x
y
y
1/x
y
Krisztina Boda
Biostat 9.*Log10 base logarithmic scale21345678910
Biostat 9.
Diagram9
0
0.3010299957
0.4771212547
0.6020599913
0.6989700043
0.7781512504
0.84509804
0.903089987
0.9542425094
1
log10 x
Munka1
z
10
21
42
83
164
1/2-1
1/4-2
1/8-3
1/16-4
Exp
xz
-20.015625-6
-1.90.0192366315-5.7
-1.80.0236830714-5.4
-1.70.029157281-5.1
-1.60.0358968236-4.8
-1.50.0441941738-4.5
-1.40.0544094102-4.2
-1.30.0669858414-3.9
-1.20.0824692444-3.6
-1.10.1015315495-3.3
-10.125-3
-0.90.1538930517-2.7
-0.80.1894645708-2.4
-0.70.2332582479-2.1
-0.60.2871745887-1.8
-0.50.3535533906-1.5
-0.40.4352752816-1.2
-0.30.5358867313-0.9
-0.20.6597539554-0.6
-0.10.8122523964-0.3
-0.090.8293195458-0.27
-0.080.8467453124-0.24
-0.070.8645372313-0.21
-0.060.8827029963-0.18
-0.050.9012504626-0.15
-0.040.9201876506-0.12
-0.030.9395227492-0.09
-0.020.9592641193-0.06
-0.010.9794202976-0.03
010
0.011.02101212570.03
0.021.04246576080.06
0.031.06437018250.09
0.041.08673486250.12
0.051.10956947210.15
0.061.13288388530.18
0.071.15668818390.21
0.081.18099266140.24
0.091.20580782770.27
0.11.23114441330.3
0.21.51571656650.6
0.31.86606598310.9
0.42.297396711.2
0.52.82842712471.5
0.63.48220225321.8
0.74.28709385012.1
0.85.27803164312.4
0.96.49801917082.7
183
1.19.84915530683.3
1.212.12573253213.6
1.314.92852786463.9
1.418.379173684.2
1.522.6274169984.5
1.627.85761802554.8
1.734.29675080125.1
1.842.22425314475.4
1.951.98415336685.7
2646
Exp
y
x
23x
Exp 10
x
log2 y
y
y
2
4
8
16
1
1/2
1/4
1/8
1/16
z
Exp2
xz
-20.000001-6
-1.90.0000019953-5.7
-1.80.0000039811-5.4
-1.70.0000079433-5.1
-1.60.0000158489-4.8
-1.50.0000316228-4.5
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-10.001-3
-0.90.0019952623-2.7
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010
0.011.07151930520.03
0.021.14815362150.06
0.031.23026877080.09
0.041.31825673860.12
0.051.41253754460.15
0.061.51356124840.18
0.071.62181009740.21
0.081.73780082870.24
0.091.86208713670.27
0.11.9952623150.3
0.23.98107170550.6
0.37.94328234720.9
0.415.84893192461.2
0.531.62277660171.5
0.663.0957344481.8
0.7125.89254117942.1
0.8251.1886431512.4
0.9501.18723362732.7
110003
1.11995.26231496893.3
1.23981.0717055353.6
1.37943.28234724283.9
1.415848.93192461114.2
1.531622.77660168384.5
1.663095.73444801954.8
1.7125892.5411794175.1
1.8251188.6431509585.4
1.9501187.2336272725.7
210000006
1/32
Exp2
y
x
103x
Munka2
x
log10 y
y
y
10
100
1000
10000
1
0.1
0.01
0.001
0.0001
z
Munka4
x
-30.1251.125-0.8750.250.0625-0.1258
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012020.5-11
0.011.00695555012.00695555010.00695555012.01391110010.503477775-1.00695555010.9930924954
0.021.01395947982.01395947980.01395947982.02791895960.5069797399-1.01395947980.9862327045
0.031.02101212572.02101212570.02101212572.04202425140.5105060629-1.02101212570.9794202976
0.041.02811382672.02811382670.02811382672.05622765330.5140569133-1.02811382670.9726549474
0.051.03526492382.03526492380.03526492382.07052984770.5176324619-1.03526492380.9659363289
0.061.04246576082.04246576080.04246576082.08493152170.5212328804-1.04246576080.9592641193
0.071.04971668362.04971668360.04971668362.09943336720.5248583418-1.04971668360.952637998
0.081.05701804062.05701804060.05701804062.11403608110.5285090203-1.05701804060.9460576467
0.091.06437018252.06437018250.06437018252.12874036490.5321850912-1.06437018250.9395227492
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0.41.31950791082.31950791080.31950791082.63901582150.6597539554-1.31950791080.7578582833
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123141-20.5
1.12.14354692513.14354692511.14354692514.28709385011.0717734625-2.14354692510.4665164958
1.22.297396713.297396711.297396714.594793421.148698355-2.297396710.4352752816
1.32.46228882673.46228882671.46228882674.92457765341.2311444133-2.46228882670.4061261982
1.42.63901582153.63901582151.63901582155.27803164311.3195079108-2.63901582150.3789291416
1.52.82842712473.82842712471.82842712475.65685424951.4142135624-2.82842712470.3535533906
1.63.0314331334.0314331332.0314331336.0628662661.5157165665-3.0314331330.3298769777
1.73.24900958544.24900958542.24900958546.49801917081.6245047927-3.24900958540.3077861033
1.83.48220225324.48220225322.48220225326.96440450641.7411011266-3.48220225320.2871745887
1.93.73213196614.73213196612.73213196617.46426393231.8660659831-3.73213196610.2679433656
245382-40.25
2.14.28709385015.28709385013.28709385018.57418770032.1435469251-4.28709385010.2332582479
2.24.594793425.594793423.594793429.189586842.29739671-4.594793420.2176376408
2.34.92457765345.92457765343.92457765349.84915530682.4622888267-4.92457765340.2030630991
2.45.27803164316.27803164314.278031643110.55606328622.6390158215-5.27803164310.1894645708
2.55.65685424956.65685424954.656854249511.3137084992.8284271247-5.65685424950.1767766953
2.66.0628662667.0628662665.06286626612.12573253213.031433133-6.0628662660.1649384888
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2.86.96440450647.96440450645.964404506413.92880901273.4822022532-6.96440450640.1435872944
2.97.46426393238.46426393236.464263932314.92852786463.7321319661-7.46426393230.1339716828
3897164-80.125
3.18.57418770039.57418770037.574187700317.14837540064.2870938501-8.57418770030.1166291239
3.29.1895868410.189586848.1895868418.379173684.59479342-9.189586840.1088188204
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3.612.125732532113.125732532111.125732532124.25146506426.062866266-12.12573253210.0824692444
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3.914.928527864615.928527864613.928527864629.85705572927.4642639323-14.92852786460.0669858414
4161715328-160.0625
1/32
Munka4
2x
2x+1
2x-1
2x
2x +1
2x -1
Munka3
2x
2x+1
2x-1
2x
2x +1
2x -1
2x
-2x
2x
2-x
2x
2x +1
yzxlog10 x
1610
20.3010
30.4771
40.6021
50.6990
60.7782
70.8451
480.9031
8390.9542
2101
1
0
4-1
-2
2-3
1-4
0-5
log10 x
xsin x2 sin x(sin x )/2sin 2xsin (x/2)
000000
0.020.01999866670.03999733340.00999933330.03998933420.0099998333
0.040.03998933420.07997866840.01999466710.0799146940.0199986667
0.060.05996400650.1199280130.02998200320.11971220730.0299955002
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1.180.92460601241.84921202480.46230300620.70441076580.5563610229
1.20.9320390861.86407817190.4660195430.67546318060.5646424734
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1.280.95801586031.91603172060.47900793010.54935543640.5971954414
1.30.96355818541.92711637080.48177909270.51550137180.6051864057
1.320.96871510011.93743020020.48435755010.4808226150.613116852
1.340.97348454171.94696908340.48674227080.44537464450.620985987
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1.380.98185353041.96370706070.49092676520.37239903940.6365371822
1.40.985449731.970899460.4927248650.33498815020.6442176872
1.420.98865176291.97730352570.49432588140.29704135130.651833771
1.440.99145834821.98291669640.49572917410.25861934970.659384672
1.460.99386836341.98773672680.49693418170.21978361220.666869635
1.480.99588084451.99176168910.49794042230.18059626790.6742879116
1.50.99749498661.99498997320.49874749330.14112000810.68163876
1.520.9987101441.9974202880.4993550720.10141798630.6889214451
1.540.99952583061.99905166120.49976291530.06155371740.6961352386
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5.96-0.3175885661-0.6351771321-0.158794283-0.60229313110.160890315
5.98-0.2985617425-0.597123485-0.1492808712-0.56988887490.1510127121
6-0.2794154982-0.5588309964-0.1397077491-0.5365729180.1411200081
6.02-0.2601574914-0.5203149829-0.1300787457-0.50239855890.1312131922
6.04-0.2407954251-0.4815908503-0.1203977126-0.46742046940.121293255
6.06-0.2213370439-0.4426740878-0.1106685219-0.43169460670.1113611887
6.08-0.2017901308-0.4035802615-0.1008950654-0.39527812480.1014179863
6.1-0.1821625043-0.3643250085-0.0910812521-0.35822928220.0914646422
6.12-0.1624620152-0.3249240304-0.0812310076-0.32060734920.0815021518
6.14-0.1426965435-0.285393087-0.0713482718-0.28247251280.0715315111
6.16-0.1228739951-0.2457479902-0.0614369976-0.24388578070.0615537174
6.18-0.1030022987-0.2060045975-0.0515011494-0.20490888330.0515697684
6.2-0.0830894028-0.1661788056-0.0415447014-0.16560417540.0415806624
6.22-0.0631432722-0.1262865445-0.0315716361-0.12603453620.0315873984
6.24-0.0431718852-0.0863437704-0.0215859426-0.08626326860.0215909757
6.26-0.02318323-0.04636646-0.011591615-0.04635399820.0115923939
6.28-0.0031853018-0.0063706036-0.0015926509-0.00637057130.0015926529
6.30.01681390050.0336278010.00840695020.0336230472-0.0084072474
6.320.03680637740.07361275490.01840318870.073562876-0.0184063069
6.340.05678413230.11356826460.02839206620.1133850199-0.0284035259
6.360.07673917430.15347834860.03836958710.1530257719-0.0383979045
6.380.09666352160.19332704330.04833176080.1924217154-0.0483884434
6.40.11654920490.23309840970.05827460240.2315098251-0.0583741434
6.420.13638826990.27277653990.0681941350.2702275685-0.0683540061
6.440.15617278150.31234556310.07808639080.3085130055-0.0783270335
6.460.17589482610.35178965220.08794741310.3463048874-0.0882922282
6.480.19554651510.39109303020.09777325760.3835427554-0.0982485937
6.50.21511998810.43023997620.1075599940.4201670368-0.1081951345
6.520.23460741590.46921483190.1173037080.4561191406-0.1181308559
6.540.2540010040.50800200790.1270005020.4913415511-0.1280547643
6.560.2732929950.546585990.13664649750.5257779199-0.1379658673
6.580.29247567240.58495134490.14623783620.5593731561-0.1478631738
6.60.31154136350.6230827270.15577068180.5920735147-0.1577456941
6.620.33048244220.66096488440.16524122110.6238266819-0.16761244
6.640.34929133230.69858266470.17464566620.6545818596-0.1774624248
6.660.36796051060.73592102110.18398025530.6842898458-0.1872946635
6.680.38648250950.7729650190.19324125480.7129031144-0.1971081729
6.70.40484992060.80969984120.20242496030.74037589-0.2069019717
6.720.42305539710.84611079430.21152769860.7666642221-0.2166750804
6.740.44109165720.88218331430.22054582860.791726055-0.2264265218
6.760.45895148640.91790297280.22947574320.8155212951-0.2361553207
6.780.47662774110.95325548230.23831387060.8380118751-0.2458605043
6.80.49411335110.98822670230.24705667560.8591618149-0.255541102
6.820.51140132241.02280264480.25570066120.878937279-0.2651961459
6.840.52848473991.05696947990.264242370.897306631-0.2748246703
6.860.54535677061.09071354130.27267838530.9142404838-0.2844257125
6.880.56201066591.12402133180.2810053330.9297117469-0.2939983124
6.90.57843976441.15687952880.28921988220.9436956694-0.3035415127
6.920.59463749471.18927498940.29731874730.9561698803-0.3130543591
6.940.61059737791.22119475580.3052986890.9671144233-0.3225359003
6.960.62631303031.25262606060.31315651520.9765117895-0.3319851882
6.980.64177816591.28355633170.32088908290.9843469452-0.3414012779
70.65698659871.31397319740.32849329940.9906073557-0.3507832277
sin x
2 sin x
(sin x )/2
sin x
sin 2x
sin (x/2)
Krisztina Boda
Biostat 9.*Logarithmic papersSemilogarithmic paperlog-log paper
Biostat 9.
Krisztina Boda
Biostat 9.*Reciprocal relationship ->take reciprocal of xModel: y=a +b/xy=a +b*1/xso y is linear in 1/x
Biostat 9.
adatok
Nameretailtheatermathlanguage
Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
0
0
0
0
0
0
0
math
math score
theater
nemlineris
0
0
0
0
0
0
math
math score
theater
transzformcik
0
0
0
0
0
0
0
math
math score
language
0
0
0
0
0
0
0
math
math score
language
0
0
0
0
0
0
math
math score
retailing
xy
-3-0.1411200081
-2.8-0.3349881502
-2.6-0.5155013718
-2.4-0.6754631806
-2.2-0.8084964038
-2-0.9092974268
-1.8-0.9738476309
-1.6-0.999573603
-1.4-0.98544973
-1.2-0.932039086
-1-0.8414709848
-0.8-0.7173560909
-0.6-0.5646424734
-0.4-0.3894183423
-0.2-0.1986693308
00
0.20.1986693308
0.40.3894183423
0.60.5646424734
0.80.7173560909
10.8414709848
1.20.932039086
1.40.98544973
1.60.999573603
1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
0
0.1986693308
0.3894183423
0.5646424734
0.7173560909
0.8414709848
0.932039086
0.98544973
0.999573603
0.9738476309
0.9092974268
0.8084964038
0.6754631806
0.5155013718
0.3349881502
y
xylg y
01.10.0413926852
11.90.278753601
240.6020599913
38.10.9084850189
4161.2041199827
xylog x
10.10
420.6020599913
83.010.903089987
163.91.2041199827
xylog xlog y
1200.3010299957
2160.30102999571.2041199827
3540.47712125471.7323937598
41280.60205999132.1072099696
xy1/x
11.11
20.450.5
30.3330.3333333333
40.230.25
50.19990.2
0
0
0
0
0
y
x
y
0
0
0
0
0
y
x
log y
0
0
0
0
y
x
y
0
0
0
0
y
log10 x
y
0
0
0
0
y
x
y
0
0
0
0
log y
log x
log y
0
0
0
0
0
y
x
y
0
0
0
0
0
y
1/x
y
Diagram13
1.1
0.45
0.333
0.23
0.1999
y
x
y
adatok
Nameretailtheatermathlanguage
Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
math
math score
theater
nemlineris
math
math score
theater
transzformcik
math
math score
language
math
math score
language
math
math score
retailing
xy
-3-0.1411200081
-2.8-0.3349881502
-2.6-0.5155013718
-2.4-0.6754631806
-2.2-0.8084964038
-2-0.9092974268
-1.8-0.9738476309
-1.6-0.999573603
-1.4-0.98544973
-1.2-0.932039086
-1-0.8414709848
-0.8-0.7173560909
-0.6-0.5646424734
-0.4-0.3894183423
-0.2-0.1986693308
00
0.20.1986693308
0.40.3894183423
0.60.5646424734
0.80.7173560909
10.8414709848
1.20.932039086
1.40.98544973
1.60.999573603
1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
0
0.1986693308
0.3894183423
0.5646424734
0.7173560909
0.8414709848
0.932039086
0.98544973
0.999573603
0.9738476309
0.9092974268
0.8084964038
0.6754631806
0.5155013718
0.3349881502
y
xylg y
01.10.0413926852
11.90.278753601
240.6020599913
38.10.9084850189
4161.2041199827
xylog x
10.10
420.6020599913
83.010.903089987
163.91.2041199827
xylog xlog y
1200.3010299957
2160.30102999571.2041199827
3540.47712125471.7323937598
41280.60205999132.1072099696
xy1/x
11.11
20.450.5
30.3330.3333333333
40.230.25
50.19990.2
y
x
y
y
x
log y
y
x
y
y
log10 x
y
y
x
y
log y
log x
log y
y
x
y
y
1/x
y
Diagram14
1.1
0.45
0.333
0.23
0.1999
y
1/x
y
adatok
Nameretailtheatermathlanguage
Pat5130525550
Sue5560515535
Inez5890510535
Amie6350495520-0.2157385322
Gene85304304550.7413007121
Bob9590400420
Joe1608004100.9989149375
-0.258397254
adatok
math
math score
theater
nemlineris
math
math score
theater
transzformcik
math
math score
language
math
math score
language
math
math score
retailing
xy
-3-0.1411200081
-2.8-0.3349881502
-2.6-0.5155013718
-2.4-0.6754631806
-2.2-0.8084964038
-2-0.9092974268
-1.8-0.9738476309
-1.6-0.999573603
-1.4-0.98544973
-1.2-0.932039086
-1-0.8414709848
-0.8-0.7173560909
-0.6-0.5646424734
-0.4-0.3894183423
-0.2-0.1986693308
00
0.20.1986693308
0.40.3894183423
0.60.5646424734
0.80.7173560909
10.8414709848
1.20.932039086
1.40.98544973
1.60.999573603
1.80.97384763090.7964355021
20.9092974268
2.20.80849640380.1574204489
2.40.6754631806
2.60.5155013718
2.80.3349881502
30.1411200081
0
0.1986693308
0.3894183423
0.5646424734
0.7173560909
0.8414709848
0.932039086
0.98544973
0.999573603
0.9738476309
0.9092974268
0.8084964038
0.6754631806
0.5155013718
0.3349881502
y
xylg y
01.10.0413926852
11.90.278753601
240.6020599913
38.10.9084850189
4161.2041199827
xylog x
10.10
420.6020599913
83.010.903089987
163.91.2041199827
xylog xlog y
1200.3010299957
2160.30102999571.2041199827
3540.47712125471.7323937598
41280.60205999132.1072099696
xy1/x
11.11
20.450.5
30.3330.3333333333
40.230.25
50.19990.2
y
x
y
y
x
log y
y
x
y
y
log10 x
y
y
x
y
log y
log x
log y
y
x
y
y
1/x
y
Krisztina Boda
Biostat 9.*Example from the literature
Biostat 9.
Krisztina Boda
Biostat 9.*
Biostat 9.
Krisztina Boda
Biostat 9.*Example 2. EL HADJ OTHMANE TAHA s mtsai: Osteoprotegerin: a regultor, a protektor s a marker. Orvosi Hetilap 2008 149. vfolyam, 42. szm 19711980.
Biostat 9.
Krisztina Boda
Biostat 9.*Useful WEB pageshttp://davidmlane.com/hyperstat/desc_biv.htmlhttp://onlinestatbook.com/stat_sim/reg_by_eye/index.htmlhttp://www.youtube.com/watch?v=CSYTZWFnVpg&feature=relatedhttp://www.statsoft.com/textbook/basic-statistics/#Correlationsbhttp://people.revoledu.com/kardi/tutorial/Regression/NonLinear/LogarithmicCurve.htmhttp://www.physics.uoguelph.ca/tutorials/GLP/
Biostat 9.
Krisztina Boda
Biostat 9.*Review questions and problemsGraphical examination of the relationship between two continuous variables (scatterplot)The meaning and properties of the coefficient of correlationCoefficient of correlation and linearityThe significance of correlation: null hypothesis, t-value, degrees of freedom, decisionThe coefficient of determinationThe meaning of the regression line and its coefficientsThe principle of finding the equation of the regression lineHypothesis test for the regressionRegression using transformations.
Biostat 9.
Krisztina Boda
Biostat 9.*ProblemsBased on n=5 pairs of observations, the coefficient of correlation was calculated, its value r=0.7. Is the correlation signficnat at 5% level? Null and alternative hypothesis:.t-value of the correlation:......................... Degrees of freedom:...................................Decision (t-value in the table t3,0.05=3.182) ..On the physics practicals the waist circumference was measured. The measurement was repeated three times. The relationship of the first two measurements were examined by linear regression. Interpret the results below ( coefficient of correlation, coefficient of determination, the significance of correlation . Null hypothesis, t-value, p-value, the equation of the regression line)
Biostat 9.
Krisztina Boda
Biostat 9.*The origin of the word regression. Galton: Regression towards mediocrity in hereditary stature. Journal of the Anthropological Institute 1886 Vol.15, 246-63
Biostat 9.
Krisztina Boda
Biostat 9.*
Biostat 9.
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