teaching statistical concepts with simulated data
DESCRIPTION
Teaching Statistical Concepts with Simulated Data. Andrej Blejec National Institute of Biology and University of Ljubljana Ljubljana, Slovenia [email protected]. Kinds of data. real life data invented data simulated data. Real life data. interesting for students - PowerPoint PPT PresentationTRANSCRIPT
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Teaching Statistical Concepts with Simulated Data
Andrej Blejec
National Institute of Biology andUniversity of LjubljanaLjubljana, Slovenia
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Kinds of data
• real life data• invented data• simulated data
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Real life data
• interesting for students• need subject matter knowledge to interpret
statistical results• oversimplified problems• complex problems• unclear relevance of statistical results
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Invented data
• a collection of data, no story• useful mostly for calculation drill• no possible interpretation of results• uninteresting for students
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Simulated data
• sampled from known distribution• composed according to prespecified model
e.g. Y = a + b * X + , with known distribution of X and
• known statistical propertiese.g. mean, variance, regression coefficient
• analysis results can be compared to true values
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Goals of teaching statistics
• interest for statistics• understanding of statistical methods• skills in use of statistics• experiences in analyzing data• other ( insert your own :) ___________
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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real life data
invented data
simulated data
interest
understanding
skills
experiences
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Sun
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
Absorption spectrum
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Sun
Hydrogen
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Sun
Hydrogen
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Least squares estimate of the mean
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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True mean value μTrue mean value μ
LS estimateLS estimate
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Maximum likelihood estimation of the mean
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-20
Log
likel
ihoo
dp
( x
| sd
)
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x
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sd)
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0.0
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sd
Log
likel
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d
0 1 2 3 4 5 6
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-60
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Log
likel
ihoo
dp
( x
| sd
)
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x
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sd)
0 1 2 3 4 5 6
0.0
0.2
0.4
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sd
Log
likel
ihoo
d
0 1 2 3 4 5 6
-100
-60
-20
Log
likel
ihoo
dp
( x
| sd
)
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x
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sd)
0 1 2 3 4 5 6
0.0
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0.4
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Log
likel
ihoo
d
0 1 2 3 4 5 6
-100
-60
-20
Log
likel
ihoo
dp
( x
| sd
)
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x
p(x|
sd)
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
sd
Log
likel
ihoo
d
0 1 2 3 4 5 6
-100
-60
-20
Log
likel
ihoo
dp
( x
| sd
)
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x
p(x|
sd)
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
sd
Log
likel
ihoo
d
0 1 2 3 4 5 6
-100
-60
-20
Log
likel
ihoo
dp
( x
| sd
)
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x
p(x|
sd)
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
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Log
likel
ihoo
d
0 1 2 3 4 5 6
-100
-60
-20
Log
likel
ihoo
dp
( x
| sd
)
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x
p(x|
sd)
0 1 2 3 4 5 6
0.0
0.2
0.4
0.6
sd
Log
likel
ihoo
d
0 1 2 3 4 5 6
-100
-60
-20
Log
likel
ihoo
dp
( x
| sd
)
ML estimateML estimate
True standard deviation σTrue standard deviation σ
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Hypothesis testing:
significance
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Hypothesis testing:
significance
or rather a surprise ?
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020
4060
8010
012
0
V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12
H0: mu= 50
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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020
4060
8010
012
0
V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12
H0: mu= 50
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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020
4060
8010
012
0
V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12
H0: mu= 50
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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020
4060
8010
012
0
V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 V12
H0: mu= 50
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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020
4060
8010
012
0
SAMPLE
H0: mu= 50
020
4060
8010
012
0
SAMPLE
H1: mu= 80
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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020
4060
8010
012
0
SAMPLE
H0: mu= 50
020
4060
8010
012
0
SAMPLE
H1: mu= 80
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Confidence intervals
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Weldon, K.L. Statistics: A Conceptual Approach.. Prentice-Hall. New York. 1986
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Bad sampling implies bias
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Bias due to “leave max out”
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Confidence intervals for variance
Biased and unbiased estimators
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Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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var
0 5 10 15 20 25
02
04
06
08
01
00
rnorm | var | n= 5 | biased
var
0 1 2 3 4 5
02
04
06
08
01
00
0 1 2 3 4 5
02
46
81
01
2
var var
0 1 2 3 4 5
02
04
06
08
01
00
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var
0 5 10 15 20 25
02
04
06
08
01
00
rnorm | var | n= 5 |
var
0 1 2 3 4 5
02
04
06
08
01
00
0 1 2 3 4 5
02
46
81
0
var var
0 1 2 3 4 5
02
04
06
08
01
00
unbiased
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var
0 2 4 6
02
04
06
08
01
00
rnorm | var | n= 10 |
var
0 1 2 3 4 5
02
04
06
08
01
00
0 1 2 3 4 5
05
10
15
var var
0 1 2 3 4 5
02
04
06
08
01
00
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var
0 1 2 3 4
02
04
06
08
01
00
rnorm | var | n= 25 |
var
0 1 2 3 4 5
02
04
06
08
01
00
0 1 2 3 4 5
05
10
15
var var
0 1 2 3 4 5
02
04
06
08
01
00
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var
0.0 0.5 1.0 1.5 2.0
02
04
06
08
01
00
rnorm | var | n= 50 |
var
0 1 2 3 4 5
02
04
06
08
01
00
0 1 2 3 4 5
05
10
15
20
25
var var
0 1 2 3 4 5
02
04
06
08
01
00
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var
0.0 0.5 1.0 1.5 2.0
02
04
06
08
01
00
rnorm | var | n= 100 |
var
0 1 2 3 4 5
02
04
06
08
01
00
0 1 2 3 4 5
01
02
03
0
var var
0 1 2 3 4 5
02
04
06
08
01
00
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var
0.0 0.5 1.0 1.5
02
04
06
08
01
00
rnorm | var | n= 200 |
var
0 1 2 3 4 5
02
04
06
08
01
00
0 1 2 3 4 5
01
02
03
04
0
var var
0 1 2 3 4 5
02
04
06
08
01
00
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Confidence intervals for variance:
what if assumptions are not met?
Case of asymetric parent population
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var
0 100 200 300 400
02
04
06
08
01
00
rlnorm | var | n= 5 |
var
0 5 10 15 20
02
04
06
08
01
00
0 5 10 15 20
01
02
03
04
0
var var
0 5 10 15 20
02
04
06
08
01
00
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var
0 50 100 150
02
04
06
08
01
00
rlnorm | var | n= 10 |
var
0 5 10 15 20
02
04
06
08
01
00
0 5 10 15 20
05
10
15
20
25
30
var var
0 5 10 15 20
02
04
06
08
01
00
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var
0 50 100 150 200
02
04
06
08
01
00
rlnorm | var | n= 25 |
var
0 5 10 15 20
02
04
06
08
01
00
0 5 10 15 20
05
10
15
20
25
30
var var
0 5 10 15 20
02
04
06
08
01
00
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var
0 10 20 30 40 50
02
04
06
08
01
00
rlnorm | var | n= 50 |
var
0 5 10 15 20
02
04
06
08
01
00
0 5 10 15 20
05
10
15
20
25
var var
0 5 10 15 20
02
04
06
08
01
00
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var
0 5 10 15 20
02
04
06
08
01
00
rlnorm | var | n= 100 |
var
0 5 10 15 20
02
04
06
08
01
00
0 5 10 15 20
05
10
15
20
var var
0 5 10 15 20
02
04
06
08
01
00
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var
0 5 10 15 20 25 30
02
04
06
08
01
00
rlnorm | var | n= 200 |
var
0 5 10 15 20
02
04
06
08
01
00
0 5 10 15 20
05
10
15
20
25
30
var var
0 5 10 15 20
02
04
06
08
01
00
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Confidence band in linear regression
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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x
y
-3 -2 -1 0 1 2 3
-3-2
-10
12
3
x
y
-3 -2 -1 0 1 2 3
-3-2
-10
12
3
y = 0 + 1 *N(0, 1 )+N(0, 1 ) n= 50 Conf. level = 0.95 r^2 = 0.5
y = 0 + 1 *N(0, 1 )+N(0, 1 ) n= 50 Conf. level = 0.95 r^2 = 0.5
y = -0.142 + 0.974 * Xr^2= 0.538
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x
y
-3 -2 -1 0 1 2 3
-3-2
-10
12
3
x
y
-3 -2 -1 0 1 2 3
-3-2
-10
12
3
y = 0 + 1 *N(0, 1 )+N(0, 1 ) n= 50 Conf. level = 0.95 r^2 = 0.5
y = 0 + 1 *N(0, 1 )+N(0, 1 ) n= 50 Conf. level = 0.95 r^2 = 0.5
y = -0.142 + 0.974 * Xr^2= 0.538
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x
y
-3 -2 -1 0 1 2 3
-3-2
-10
12
3
x
y
-3 -2 -1 0 1 2 3
-3-2
-10
12
3
y = 0 + 1 *N(0, 1 )+N(0, 1 ) n= 50 Conf. level = 0.95 r^2 = 0.5
y = 0 + 1 *N(0, 1 )+N(0, 1 ) n= 50 Conf. level = 0.95 r^2 = 0.5
0.153 + 1.07 * Xr^2= 0.413
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x
y
-3 -2 -1 0 1 2 3
-3-2
-10
12
3y = 0 + 1 *N(0, 1 )+N(0, 1 )
n= 40 Conf. level = 0.95 r^2 = 0.5
-0.0934 + 0.923 * X
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x
y
-3 -2 -1 0 1 2 3
-3-2
-10
12
3y = 0 + 1 *N(0, 1 )+N(0, 1 )
n= 40 Conf. level = 0.95 r^2 = 0.5
-0.0934 + 0.923 * X-0.0934 + 0.923 * X-0.11 + 1.04 * X-0.11 + 1.04 * X-0.00754 + 1.01 * X-0.00754 + 1.01 * X0.0667 + 1.23 * X
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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x
y
-3 -2 -1 0 1 2 3
-3-2
-10
12
3y = 0 + 1 *N(0, 1 )+N(0, 1 )
n= 40 Conf. level = 0.95 r^2 = 0.5
-0.0934 + 0.923 * X-0.0934 + 0.923 * X-0.11 + 1.04 * X-0.11 + 1.04 * X-0.00754 + 1.01 * X-0.00754 + 1.01 * X0.0667 + 1.23 * X0.0667 + 1.23 * X-0.0458 + 1.18 * X-0.0458 + 1.18 * X0.0297 + 1.08 * X0.0297 + 1.08 * X-0.205 + 0.785 * X-0.205 + 0.785 * X0.0115 + 0.833 * X0.0115 + 0.833 * X0.0933 + 1.2 * X0.0933 + 1.2 * X0.00637 + 0.968 * X0.00637 + 0.968 * X
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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x
y
-3 -2 -1 0 1 2 3
-3-2
-10
12
3y = 0 + 1 *N(0, 1 )+N(0, 1 )
n= 40 Conf. level = 0.95 r^2 = 0.5
-0.2 + 0.841 * X-0.2 + 0.841 * X-0.257 + 1.18 * X-0.257 + 1.18 * X0.116 + 0.751 * X0.116 + 0.751 * X0.00216 + 0.907 * X
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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x
y
-3 -2 -1 0 1 2 3
-3-2
-10
12
3y = 0 + 1 *N(0, 1 )+N(0, 1 )
n= 40 Conf. level = 0.95 r^2 = 0.5
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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x
y
-3 -2 -1 0 1 2 3
-3-2
-10
12
3y = 0 + 1 *N(0, 1 )+N(0, 4 )
n= 40 Conf. level = 0.95 r^2 = 0.2
-0.4 + 0.683 * X
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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x
y
-3 -2 -1 0 1 2 3
-3-2
-10
12
3y = 0 + 1 *N(0, 1 )+N(0, 4 )
n= 40 Conf. level = 0.95 r^2 = 0.2
-0.4 + 0.683 * X-0.4 + 0.683 * X-0.514 + 1.36 * X-0.514 + 1.36 * X0.232 + 0.501 * X0.232 + 0.501 * X0.00432 + 0.815 * X0.00432 + 0.815 * X-0.695 + 1.04 * X-0.695 + 1.04 * X-0.0815 + 0.528 * X-0.0815 + 0.528 * X-0.234 + 1.07 * X-0.234 + 1.07 * X-0.723 + 0.653 * X-0.723 + 0.653 * X0.161 + 1.56 * X0.161 + 1.56 * X-0.359 + 0.595 * X-0.359 + 0.595 * X
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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x
y
-3 -2 -1 0 1 2 3
-3-2
-10
12
3y = 0 + 1 *N(0, 1 )+N(0, 4 )
n= 40 Conf. level = 0.95 r^2 = 0.2
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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Final remarks
• Graphically supported simulations can to some extent ‑ ‑replace the proofs, usually not understandable to non-mathematics majors. Maybe they are the answer to some questions:
• "If an audience is not convinced by proof, why do proof? (Moore, 1996)
• “Do [students] need to know the theory or do they need to understand the concept?” (J Brown and I David, ICOTS8, 2010)
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011
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• Simulated data have to be combined with real life data and projects
• Simulated data can serve as pure and simple data on which we can train our perception for statistical results and learn what patterns and properties in data can be revealed by applied method
• After such preparation students will be able to interpret the real life and subject matter data in all their complexity
Workshop in Methodology of Teaching Statistics / Novi Sad, December 13, 2011