Mixed model help

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Mixed model help

Chris Hunt-2

Hi,

 

Can anybody help explain why, when I run the following mixed model on the data below, I get different ‘denominator df’ for the fixed effects for var1 and var2?

For Var1 all the fixed effects have a denominator df of 32 and for Var2 I get three different dfs none of which are whole numbers.  I could understand this if there was missing data but there is none and all subjects have completed all 5 parts of the study.

 

Any help in understanding this would be great.

 

Many thanks

Chris

 

 

 

MIXED Var1 BY sub order type

  /FIXED=order type order*type | NOINT

  /RANDOM=sub | COVTYPE(ID).

 

 

 

sub

order

type

Var1

Var2

1

1

5

-0.2

-2

1

2

2

1.0

32

1

3

3

1.0

80

1

4

4

0.0

0

1

5

1

0.6

18

2

1

1

1.4

44

2

2

2

-0.4

-4

2

3

4

1.2

52

2

4

5

-0.2

0

2

5

3

1.0

78

3

1

5

0.2

2

3

2

4

0.0

0

3

3

3

1.0

82

3

4

2

1.2

52

3

5

1

0.8

62

5

1

5

0.0

0

5

2

1

0.6

28

5

3

2

1.2

72

5

4

3

1.2

80

5

5

4

0.0

0

6

1

3

1.0

82

6

2

2

1.0

72

6

3

5

0.0

0

6

4

1

1.0

40

6

5

4

0.0

0

7

1

1

0.2

6

7

2

3

1.0

78

7

3

5

0.0

0

7

4

2

0.8

18

7

5

4

0.2

2

8

1

3

1.2

82

8

2

4

0.0

0

8

3

2

1.0

14

8

4

5

0.6

2

8

5

1

1.2

60

9

1

4

0.0

0

9

2

3

1.8

26

9

3

1

0.2

2

9

4

2

0.2

2

9

5

5

0.0

0

10

1

3

1.0

80

10

2

1

1.0

54

10

3

2

0.8

38

10

4

5

0.0

0

10

5

4

0.0

0

11

1

3

0.8

48

11

2

2

1.0

24

11

3

4

0.2

0

11

4

5

0.0

0

11

5

1

1.0

56

12

1

5

0.2

2

12

2

1

0.0

0

12

3

3

1.0

72

12

4

4

0.0

0

12

5

2

1.0

34

13

1

2

0.8

56

13

2

3

1.0

80

13

3

1

1.0

36

13

4

4

0.0

0

13

5

5

0.4

8

 

 

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Re: Mixed model help

charla3@aol.com
Chris - Why would you include subject as an independent variable in this model?   Did you try this:

MIXED Var1 BY order type
  /FIXED=order type order*type | NOINT
  /RANDOM=sub | COVTYPE(ID).
 



-----Original Message-----
From: Chris Hunt <[hidden email]>
To: [hidden email]
Sent: Thu, Feb 11, 2010 4:42 am
Subject: Mixed model help

Hi,
 
Can anybody help explain why, when I run the following mixed model on the data below, I get different ‘denominator df’ for the fixed effects for var1 and var2?
For Var1 all the fixed effects have a denominator df of 32 and for Var2 I get three different dfs none of which are whole numbers.  I could understand this if there was missing data but there is none and all subjects have completed all 5 parts of the study.
 
Any help in understanding this would be great.
 
Many thanks
Chris
 
 
 
MIXED Var1 BY sub order type
  /FIXED=order type order*type | NOINT
  /RANDOM=sub | COVTYPE(ID).
 
 
 
sub
order
type
Var1
Var2
1
1
5
-0.2
-2
1
2
2
1.0
32
1
3
3
1.0
80
1
4
4
0.0
0
1
5
1
0.6
18
2
1
1
1.4
44
2
2
2
-0.4
-4
2
3
4
1.2
52
2
4
5
-0.2
0
2
5
3
1.0
78
3
1
5
0.2
2
3
2
4
0.0
0
3
3
3
1.0
82
3
4
2
1.2
52
3
5
1
0.8
62
5
1
5
0.0
0
5
2
1
0.6
28
5
3
2
1.2
72
5
4
3
1.2
80
5
5
4
0.0
0
6
1
3
1.0
82
6
2
2
1.0
72
6
3
5
0.0
0
6
4
1
1.0
40
6
5
4
0.0
0
7
1
1
0.2
6
7
2
3
1.0
78
7
3
5
0.0
0
7
4
2
0.8
18
7
5
4
0.2
2
8
1
3
1.2
82
8
2
4
0.0
0
8
3
2
1.0
14
8
4
5
0.6
2
8
5
1
1.2
60
9
1
4
0.0
0
9
2
3
1.8
26
9
3
1
0.2
2
9
4
2
0.2
2
9
5
5
0.0
0
10
1
3
1.0
80
10
2
1
1.0
54
10
3
2
0.8
38
10
4
5
0.0
0
10
5
4
0.0
0
11
1
3
0.8
48
11
2
2
1.0
24
11
3
4
0.2
0
11
4
5
0.0
0
11
5
1
1.0
56
12
1
5
0.2
2
12
2
1
0.0
0
12
3
3
1.0
72
12
4
4
0.0
0
12
5
2
1.0
34
13
1
2
0.8
56
13
2
3
1.0
80
13
3
1
1.0
36
13
4
4
0.0
0
13
5
5
0.4
8
 
 
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Re: Mixed model help

Ryan
In reply to this post by Chris Hunt-2
Chris,

Although I typically do not code up these types of models this way, I do think the code *could be* correct. There is one glaring issue, though...Why are you excluding the fixed intercept from the equation? Perhaps you can give us some context?

Ryan

p.s. I am assuming that you have met the assumptions to run this type of analysis.

Chris Hunt-2 wrote
Hi,

 

Can anybody help explain why, when I run the following mixed model on
the data below, I get different 'denominator df' for the fixed effects
for var1 and var2?

For Var1 all the fixed effects have a denominator df of 32 and for Var2
I get three different dfs none of which are whole numbers.  I could
understand this if there was missing data but there is none and all
subjects have completed all 5 parts of the study.

 

Any help in understanding this would be great.

 

Many thanks

Chris

 

 

 

MIXED Var1 BY sub order type

  /FIXED=order type order*type | NOINT

  /RANDOM=sub | COVTYPE(ID).

 

 

 

sub

order

type

Var1

Var2

1

1

5

-0.2

-2

1

2

2

1.0

32

1

3

3

1.0

80

1

4

4

0.0

0

1

5

1

0.6

18

2

1

1

1.4

44

2

2

2

-0.4

-4

2

3

4

1.2

52

2

4

5

-0.2

0

2

5

3

1.0

78

3

1

5

0.2

2

3

2

4

0.0

0

3

3

3

1.0

82

3

4

2

1.2

52

3

5

1

0.8

62

5

1

5

0.0

0

5

2

1

0.6

28

5

3

2

1.2

72

5

4

3

1.2

80

5

5

4

0.0

0

6

1

3

1.0

82

6

2

2

1.0

72

6

3

5

0.0

0

6

4

1

1.0

40

6

5

4

0.0

0

7

1

1

0.2

6

7

2

3

1.0

78

7

3

5

0.0

0

7

4

2

0.8

18

7

5

4

0.2

2

8

1

3

1.2

82

8

2

4

0.0

0

8

3

2

1.0

14

8

4

5

0.6

2

8

5

1

1.2

60

9

1

4

0.0

0

9

2

3

1.8

26

9

3

1

0.2

2

9

4

2

0.2

2

9

5

5

0.0

0

10

1

3

1.0

80

10

2

1

1.0

54

10

3

2

0.8

38

10

4

5

0.0

0

10

5

4

0.0

0

11

1

3

0.8

48

11

2

2

1.0

24

11

3

4

0.2

0

11

4

5

0.0

0

11

5

1

1.0

56

12

1

5

0.2

2

12

2

1

0.0

0

12

3

3

1.0

72

12

4

4

0.0

0

12

5

2

1.0

34

13

1

2

0.8

56

13

2

3

1.0

80

13

3

1

1.0

36

13

4

4

0.0

0

13

5

5

0.4

8

 

 
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Re: Mixed model help

Chris Hunt
Thanks for your reply,
The reason for not including the intercept is that whether it is in or not
does not change the analysis.  So as I was not going to report it I thought
it easier to leave it out.

The data is from a 5x5 cross-over study.

I have modified the model by nesting the random effect subject in order.
This gives a better looking result.

Yes I do believe all the assumption are met. Below was just a sample of my
sample!

Chris



-----Original Message-----
From: SPSSX(r) Discussion [mailto:[hidden email]] On Behalf Of
rblack
Sent: 12 February 2010 03:03
To: [hidden email]
Subject: Re: Mixed model help

Chris,

Although I typically do not code up these types of models this way, I do
think the code *could be* correct. There is one glaring issue, though...Why
are you excluding the fixed intercept from the equation? Perhaps you can
give us some context?

Ryan

p.s. I am assuming that you have met the assumptions to run this type of
analysis.


Chris Hunt-2 wrote:

>
> Hi,
>
>
>
> Can anybody help explain why, when I run the following mixed model on
> the data below, I get different 'denominator df' for the fixed effects
> for var1 and var2?
>
> For Var1 all the fixed effects have a denominator df of 32 and for Var2
> I get three different dfs none of which are whole numbers.  I could
> understand this if there was missing data but there is none and all
> subjects have completed all 5 parts of the study.
>
>
>
> Any help in understanding this would be great.
>
>
>
> Many thanks
>
> Chris
>
>
>
>
>
>
>
> MIXED Var1 BY sub order type
>
>   /FIXED=order type order*type | NOINT
>
>   /RANDOM=sub | COVTYPE(ID).
>
>
>
>
>
>
>
> sub
>
> order
>
> type
>
> Var1
>
> Var2
>
> 1
>
> 1
>
> 5
>
> -0.2
>
> -2
>
> 1
>
> 2
>
> 2
>
> 1.0
>
> 32
>
> 1
>
> 3
>
> 3
>
> 1.0
>
> 80
>
> 1
>
> 4
>
> 4
>
> 0.0
>
> 0
>
> 1
>
> 5
>
> 1
>
> 0.6
>
> 18
>
> 2
>
> 1
>
> 1
>
> 1.4
>
> 44
>
> 2
>
> 2
>
> 2
>
> -0.4
>
> -4
>
> 2
>
> 3
>
> 4
>
> 1.2
>
> 52
>
> 2
>
> 4
>
> 5
>
> -0.2
>
> 0
>
> 2
>
> 5
>
> 3
>
> 1.0
>
> 78
>
> 3
>
> 1
>
> 5
>
> 0.2
>
> 2
>
> 3
>
> 2
>
> 4
>
> 0.0
>
> 0
>
> 3
>
> 3
>
> 3
>
> 1.0
>
> 82
>
> 3
>
> 4
>
> 2
>
> 1.2
>
> 52
>
> 3
>
> 5
>
> 1
>
> 0.8
>
> 62
>
> 5
>
> 1
>
> 5
>
> 0.0
>
> 0
>
> 5
>
> 2
>
> 1
>
> 0.6
>
> 28
>
> 5
>
> 3
>
> 2
>
> 1.2
>
> 72
>
> 5
>
> 4
>
> 3
>
> 1.2
>
> 80
>
> 5
>
> 5
>
> 4
>
> 0.0
>
> 0
>
> 6
>
> 1
>
> 3
>
> 1.0
>
> 82
>
> 6
>
> 2
>
> 2
>
> 1.0
>
> 72
>
> 6
>
> 3
>
> 5
>
> 0.0
>
> 0
>
> 6
>
> 4
>
> 1
>
> 1.0
>
> 40
>
> 6
>
> 5
>
> 4
>
> 0.0
>
> 0
>
> 7
>
> 1
>
> 1
>
> 0.2
>
> 6
>
> 7
>
> 2
>
> 3
>
> 1.0
>
> 78
>
> 7
>
> 3
>
> 5
>
> 0.0
>
> 0
>
> 7
>
> 4
>
> 2
>
> 0.8
>
> 18
>
> 7
>
> 5
>
> 4
>
> 0.2
>
> 2
>
> 8
>
> 1
>
> 3
>
> 1.2
>
> 82
>
> 8
>
> 2
>
> 4
>
> 0.0
>
> 0
>
> 8
>
> 3
>
> 2
>
> 1.0
>
> 14
>
> 8
>
> 4
>
> 5
>
> 0.6
>
> 2
>
> 8
>
> 5
>
> 1
>
> 1.2
>
> 60
>
> 9
>
> 1
>
> 4
>
> 0.0
>
> 0
>
> 9
>
> 2
>
> 3
>
> 1.8
>
> 26
>
> 9
>
> 3
>
> 1
>
> 0.2
>
> 2
>
> 9
>
> 4
>
> 2
>
> 0.2
>
> 2
>
> 9
>
> 5
>
> 5
>
> 0.0
>
> 0
>
> 10
>
> 1
>
> 3
>
> 1.0
>
> 80
>
> 10
>
> 2
>
> 1
>
> 1.0
>
> 54
>
> 10
>
> 3
>
> 2
>
> 0.8
>
> 38
>
> 10
>
> 4
>
> 5
>
> 0.0
>
> 0
>
> 10
>
> 5
>
> 4
>
> 0.0
>
> 0
>
> 11
>
> 1
>
> 3
>
> 0.8
>
> 48
>
> 11
>
> 2
>
> 2
>
> 1.0
>
> 24
>
> 11
>
> 3
>
> 4
>
> 0.2
>
> 0
>
> 11
>
> 4
>
> 5
>
> 0.0
>
> 0
>
> 11
>
> 5
>
> 1
>
> 1.0
>
> 56
>
> 12
>
> 1
>
> 5
>
> 0.2
>
> 2
>
> 12
>
> 2
>
> 1
>
> 0.0
>
> 0
>
> 12
>
> 3
>
> 3
>
> 1.0
>
> 72
>
> 12
>
> 4
>
> 4
>
> 0.0
>
> 0
>
> 12
>
> 5
>
> 2
>
> 1.0
>
> 34
>
> 13
>
> 1
>
> 2
>
> 0.8
>
> 56
>
> 13
>
> 2
>
> 3
>
> 1.0
>
> 80
>
> 13
>
> 3
>
> 1
>
> 1.0
>
> 36
>
> 13
>
> 4
>
> 4
>
> 0.0
>
> 0
>
> 13
>
> 5
>
> 5
>
> 0.4
>
> 8
>
>
>
>
>
>
>

--
View this message in context:
http://old.nabble.com/Mixed-model-help-tp27549169p27558321.html
Sent from the SPSSX Discussion mailing list archive at Nabble.com.

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Re: Mixed model help

Ryan
Chris,

Rarely do I exclude the fixed grand intercept from a linear model. One exception is when I want to parameterize a model in order to observe multiple intercepts and slopes in the Fixed Effects table.  See my recent post on how to run a multivariate model in the MIXED procedure for further details. Anyway, unless you have good reason to exclude the fixed grand intercept, I suggest you leave it in. As an aside, particular interest in a cross-over design, I think, would be to start by testing whether or not there is an interaction between treatment and period.

I'm not sure what to make of the degrees of freedom issue. Are you absolutely 100% positive that you do not have any missing data for the second dependent variable? I urge you to not only look for blank cells, but cells that may contain values that have been deemed MISSING. Simply compute a frequency distribution table to see if there are any missing data. I assume you have done this but I figured it was worth mentioning. Anyway, I'll keep thinking about this issue and will let you know if I come up with any other thoughts. In case you are curious, another parameterization for  the model which I would likely use for this situation would be:

MIXED Var1 BY order type
  /FIXED=order type order*type | SSTYPE(3)
  /METHOD=REML
  /RANDOM=INTERCEPT | SUBJECT(sub) COVTYPE(VC).

I would expect results from the code above to yield identical results to the code you presented in your original post. I'd be interested in hearing if that is not the case.

Ryan

Chris Hunt-3 wrote
Thanks for your reply,
The reason for not including the intercept is that whether it is in or not
does not change the analysis.  So as I was not going to report it I thought
it easier to leave it out.

The data is from a 5x5 cross-over study.

I have modified the model by nesting the random effect subject in order.
This gives a better looking result.

Yes I do believe all the assumption are met. Below was just a sample of my
sample!

Chris



-----Original Message-----
From: SPSSX(r) Discussion [mailto:SPSSX-L@LISTSERV.UGA.EDU] On Behalf Of
rblack
Sent: 12 February 2010 03:03
To: SPSSX-L@LISTSERV.UGA.EDU
Subject: Re: Mixed model help

Chris,

Although I typically do not code up these types of models this way, I do
think the code *could be* correct. There is one glaring issue, though...Why
are you excluding the fixed intercept from the equation? Perhaps you can
give us some context?

Ryan

p.s. I am assuming that you have met the assumptions to run this type of
analysis.


Chris Hunt-2 wrote:
>
> Hi,
>
>
>
> Can anybody help explain why, when I run the following mixed model on
> the data below, I get different 'denominator df' for the fixed effects
> for var1 and var2?
>
> For Var1 all the fixed effects have a denominator df of 32 and for Var2
> I get three different dfs none of which are whole numbers.  I could
> understand this if there was missing data but there is none and all
> subjects have completed all 5 parts of the study.
>
>
>
> Any help in understanding this would be great.
>
>
>
> Many thanks
>
> Chris
>
>
>
>
>
>
>
> MIXED Var1 BY sub order type
>
>   /FIXED=order type order*type | NOINT
>
>   /RANDOM=sub | COVTYPE(ID).
>
>
>
>
>
>
>
> sub
>
> order
>
> type
>
> Var1
>
> Var2
>
> 1
>
> 1
>
> 5
>
> -0.2
>
> -2
>
> 1
>
> 2
>
> 2
>
> 1.0
>
> 32
>
> 1
>
> 3
>
> 3
>
> 1.0
>
> 80
>
> 1
>
> 4
>
> 4
>
> 0.0
>
> 0
>
> 1
>
> 5
>
> 1
>
> 0.6
>
> 18
>
> 2
>
> 1
>
> 1
>
> 1.4
>
> 44
>
> 2
>
> 2
>
> 2
>
> -0.4
>
> -4
>
> 2
>
> 3
>
> 4
>
> 1.2
>
> 52
>
> 2
>
> 4
>
> 5
>
> -0.2
>
> 0
>
> 2
>
> 5
>
> 3
>
> 1.0
>
> 78
>
> 3
>
> 1
>
> 5
>
> 0.2
>
> 2
>
> 3
>
> 2
>
> 4
>
> 0.0
>
> 0
>
> 3
>
> 3
>
> 3
>
> 1.0
>
> 82
>
> 3
>
> 4
>
> 2
>
> 1.2
>
> 52
>
> 3
>
> 5
>
> 1
>
> 0.8
>
> 62
>
> 5
>
> 1
>
> 5
>
> 0.0
>
> 0
>
> 5
>
> 2
>
> 1
>
> 0.6
>
> 28
>
> 5
>
> 3
>
> 2
>
> 1.2
>
> 72
>
> 5
>
> 4
>
> 3
>
> 1.2
>
> 80
>
> 5
>
> 5
>
> 4
>
> 0.0
>
> 0
>
> 6
>
> 1
>
> 3
>
> 1.0
>
> 82
>
> 6
>
> 2
>
> 2
>
> 1.0
>
> 72
>
> 6
>
> 3
>
> 5
>
> 0.0
>
> 0
>
> 6
>
> 4
>
> 1
>
> 1.0
>
> 40
>
> 6
>
> 5
>
> 4
>
> 0.0
>
> 0
>
> 7
>
> 1
>
> 1
>
> 0.2
>
> 6
>
> 7
>
> 2
>
> 3
>
> 1.0
>
> 78
>
> 7
>
> 3
>
> 5
>
> 0.0
>
> 0
>
> 7
>
> 4
>
> 2
>
> 0.8
>
> 18
>
> 7
>
> 5
>
> 4
>
> 0.2
>
> 2
>
> 8
>
> 1
>
> 3
>
> 1.2
>
> 82
>
> 8
>
> 2
>
> 4
>
> 0.0
>
> 0
>
> 8
>
> 3
>
> 2
>
> 1.0
>
> 14
>
> 8
>
> 4
>
> 5
>
> 0.6
>
> 2
>
> 8
>
> 5
>
> 1
>
> 1.2
>
> 60
>
> 9
>
> 1
>
> 4
>
> 0.0
>
> 0
>
> 9
>
> 2
>
> 3
>
> 1.8
>
> 26
>
> 9
>
> 3
>
> 1
>
> 0.2
>
> 2
>
> 9
>
> 4
>
> 2
>
> 0.2
>
> 2
>
> 9
>
> 5
>
> 5
>
> 0.0
>
> 0
>
> 10
>
> 1
>
> 3
>
> 1.0
>
> 80
>
> 10
>
> 2
>
> 1
>
> 1.0
>
> 54
>
> 10
>
> 3
>
> 2
>
> 0.8
>
> 38
>
> 10
>
> 4
>
> 5
>
> 0.0
>
> 0
>
> 10
>
> 5
>
> 4
>
> 0.0
>
> 0
>
> 11
>
> 1
>
> 3
>
> 0.8
>
> 48
>
> 11
>
> 2
>
> 2
>
> 1.0
>
> 24
>
> 11
>
> 3
>
> 4
>
> 0.2
>
> 0
>
> 11
>
> 4
>
> 5
>
> 0.0
>
> 0
>
> 11
>
> 5
>
> 1
>
> 1.0
>
> 56
>
> 12
>
> 1
>
> 5
>
> 0.2
>
> 2
>
> 12
>
> 2
>
> 1
>
> 0.0
>
> 0
>
> 12
>
> 3
>
> 3
>
> 1.0
>
> 72
>
> 12
>
> 4
>
> 4
>
> 0.0
>
> 0
>
> 12
>
> 5
>
> 2
>
> 1.0
>
> 34
>
> 13
>
> 1
>
> 2
>
> 0.8
>
> 56
>
> 13
>
> 2
>
> 3
>
> 1.0
>
> 80
>
> 13
>
> 3
>
> 1
>
> 1.0
>
> 36
>
> 13
>
> 4
>
> 4
>
> 0.0
>
> 0
>
> 13
>
> 5
>
> 5
>
> 0.4
>
> 8
>
>
>
>
>
>
>

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Re: Mixed model help

Alex Reutter
In reply to this post by Chris Hunt-2

When you run the model on Var1 using the full dataset, do you get a warning message like:

"The final Hessian matrix is not positive definite although all convergence criteria are satisfied. The MIXED procedure continues despite this warning. Validity of subsequent results cannot be ascertained."

I get this when running on the sample.

Alex



From: Chris Hunt <[hidden email]>
To: [hidden email]
Date: 02/11/2010 09:33 AM
Subject: Mixed model help
Sent by: "SPSSX(r) Discussion" <[hidden email]>





Hi,
 
Can anybody help explain why, when I run the following mixed model on the data below, I get different ‘denominator df’ for the fixed effects for var1 and var2?
For Var1 all the fixed effects have a denominator df of 32 and for Var2 I get three different dfs none of which are whole numbers.  I could understand this if there was missing data but there is none and all subjects have completed all 5 parts of the study.
 
Any help in understanding this would be great.
 
Many thanks
Chris
 
 
 
MIXED Var1 BY sub order type
  /FIXED=order type order*type | NOINT
  /RANDOM=sub | COVTYPE(ID).
 
 
 

sub
order
type
Var1
Var2
1
1
5
-0.2
-2
1
2
2
1.0
32
1
3
3
1.0
80
1
4
4
0.0
0
1
5
1
0.6
18
2
1
1
1.4
44
2
2
2
-0.4
-4
2
3
4
1.2
52
2
4
5
-0.2
0
2
5
3
1.0
78
3
1
5
0.2
2
3
2
4
0.0
0
3
3
3
1.0
82
3
4
2
1.2
52
3
5
1
0.8
62
5
1
5
0.0
0
5
2
1
0.6
28
5
3
2
1.2
72
5
4
3
1.2
80
5
5
4
0.0
0
6
1
3
1.0
82
6
2
2
1.0
72
6
3
5
0.0
0
6
4
1
1.0
40
6
5
4
0.0
0
7
1
1
0.2
6
7
2
3
1.0
78
7
3
5
0.0
0
7
4
2
0.8
18
7
5
4
0.2
2
8
1
3
1.2
82
8
2
4
0.0
0
8
3
2
1.0
14
8
4
5
0.6
2
8
5
1
1.2
60
9
1
4
0.0
0
9
2
3
1.8
26
9
3
1
0.2
2
9
4
2
0.2
2
9
5
5
0.0
0
10
1
3
1.0
80
10
2
1
1.0
54
10
3
2
0.8
38
10
4
5
0.0
0
10
5
4
0.0
0
11
1
3
0.8
48
11
2
2
1.0
24
11
3
4
0.2
0
11
4
5
0.0
0
11
5
1
1.0
56
12
1
5
0.2
2
12
2
1
0.0
0
12
3
3
1.0
72
12
4
4
0.0
0
12
5
2
1.0
34
13
1
2
0.8
56
13
2
3
1.0
80
13
3
1
1.0
36
13
4
4
0.0
0
13
5
5
0.4
8