Maximum slope of learning curve

[Michael DeLong]

Hello!

I have three parameters that measure an examiners performance over time. The data is noted for around 10 examiners and decreases over time, indicating better performance - I already performed analysis with linear mixed model analysis.
Now I am trying to find out where the maximum slope of the learning curve lies - to get an indication on the number of necessary interventions.
Any ideas how to calculate that?

Thanks!
Michael

[Carlo Lazzaro]

Michael:
turning points concerning non-linear relationships between regressor(s) and regressand are usually investigated via squared terms, as you can see from the folowing toy-example (that does not show any evidence of non-linear relationship):

. use "http://www.stata-press.com/data/r16/pig.dta"
(Longitudinal analysis of pig weights)

. mixed weight c.week##c.week || id:

Performing EM optimization:

Performing gradient-based optimization:

Iteration 0:   log likelihood = -1014.5524 
Iteration 1:   log likelihood = -1014.5524 

Computing standard errors:

Mixed-effects ML regression                     Number of obs     =        432
Group variable: id                              Number of groups  =         48

                                                Obs per group:
                                                              min =          9
                                                              avg =        9.0
                                                              max =          9

                                                Wald chi2(2)      =   25387.69
Log likelihood = -1014.5524                     Prob > chi2       =     0.0000

-------------------------------------------------------------------------------
       weight |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
         week |   6.358818   .1763801    36.05   0.000     6.013119    6.704517
              |
c.week#c.week |  -.0148922    .017202    -0.87   0.387    -.0486076    .0188231
              |
        _cons |   19.08259   .6754836    28.25   0.000     17.75867    20.40651
-------------------------------------------------------------------------------

------------------------------------------------------------------------------
  Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
-----------------------------+------------------------------------------------
id: Identity                 |
                  var(_cons) |   14.81846   3.124225      9.802604    22.40086
-----------------------------+------------------------------------------------
               var(Residual) |   4.374725   .3157186        3.7977    5.039425
------------------------------------------------------------------------------
LR test vs. linear model: chibar2(01) = 473.23        Prob >= chibar2 = 0.0000

.

[Nick Cox]

Michael asked about the maximum slope of the learning curve. For panel data time series operators give you differences, and if data are equally spaced in time that lets you find the steepest part of each curve. If not equally spaced, correct for the length of time intervals.

[Michael DeLong]

Date is equally spaced in time, e.g. I have a continuous variable IDofIntervention. It ranges from 60 to 130 interventions and I would ideally find out something like that after performing 20.3 interventions there is a turning point where the degree of additional experience per performed intervention shrinks again.

[Nick Cox]

Sorry, but this does not get less confusing. In #1 you state

The data ... decreases over time.

which to me is consistent with wanting to know

where the maximum slope of the learning curve lies

Now in #4 you want to know

a turning point where the degree of additional experience per performed intervention shrinks again

A turning point is a maximum or minimum of a curve. The idea from calculus is that at such points the gradient or slope is zero.

The point of maximum slope in contrast will be somewhere on a rising section or a falling section of any curve with one or more turning points.

So, which is it?

Please consult https://www.statalist.org/forums/help#stata to find out how to give us a data example -- real or realistic data for one examiner, say -- and make plain what you want to calculate.

[Michael DeLong]

Sorry for the confusion, actually I am still wondering what the best parameter might be.
Below is my sample code for two interventionalists. The higher the experience, the lower the AmountOfDose should become.
So what I am trying to find out is the average NrOfIntervention needed until there is no significant gain in experience - measured as reduction of AmountOfDose per additional intervention - any more.

* Example generated by -dataex-. To install: ssc install dataex
clear
input long Interventionalist float NrOfIntervention double AmountOfDose
2   1        63
2   2      54.5
2   3        93
2   4       111
2   5       202
2   6        72
2   7        85
2   8       136
2   9        72
2  10       175
2  11        59
2  12       192
2  13       122
2  14        45
2  15        65
2  16        58
2  17        95
2  18        76
2  19       238
2  20        35
2  21        67
2  22        50
2  23        84
2  24        64
2  25        81
2  26        69
2  27       219
2  28        39
2  29        52
2  30        59
2  31       110
2  32        88
2  33        88
2  34        70
2  35        73
2  36       128
2  37        58
2  38        43
2  39        71
2  40       103
2  41        50
2  42        53
2  43      27.5
2  44        33
2  45        86
2  46        70
2  47        61
2  48        81
2  49      54.5
2  50        79
2  51        24
2  52      51.5
2  53       219
2  54        33
2  55        38
2  56       118
2  57        49
2  58        61
2  59        48
2  60      19.5
2  61        85
2  62      40.5
2  63        39
2  64        62
2  65        28
2  66        30
2  67        37
2  68        92
2  69        35
2  70        28
2  71        48
2  72        53
2  73  47.28571
2  74        59
2  75        63
2  76        47
2  77       130
2  78        48
2  79       173
2  80        44
2  81        62
2  82        87
2  83      64.5
2  84        63
2  85        43
2  86        52
2  87        54
2  88        24
2  89        33
2  90        34
2  91        76
2  92        65
2  93        74
2  94        36
2  95        39
2  96        20
2  97        28
2  98        60
2  99        56
2 100        55
2 101        37
2 102        26
2 103        44
2 104        26
2 105        55
2 106        22
2 107        15
2 108        22
2 109 10.333333
2 110        23
2 111         7
2 112        14
2 113        25
2 114        17
2 115        18
2 116         9
2 117        42
2 118       150
3   1        23
3   2         8
3   3        11
3   4        50
3   5        57
3   6        30
3   7         5
3   8        27
3   9         9
3  10        11
3  11        10
3  12        19
3  13        10
3  14        25
3  15        41
3  16        21
3  17         9
3  18        13
3  19        19
3  20        18
3  21        23
3  22         8
3  23        10
3  24        19
3  25        13
3  26         6
3  27         9
3  28         5
3  29        14
3  30      16.5
3  31         8
3  32         8
3  33         5
3  34        10
3  35         9
3  36         6
3  37         6
3  38      10.5
3  39        16
3  40        29
3  41         8
3  42         6
3  43         7
3  44         5
3  45        25
3  46         9
3  47         3
3  48         4
3  49         4
3  50         4
3  51         4
3  52         8
3  53        43
3  54         7
3  55        18
3  56         5
3  57         4
3  58         6
3  59        13
3  60         9
3  61         6
3  62         5
3  63      12.5
3  64         7
3  65         4
3  66        10
3  67        12
3  68         7
3  69         5
3  70         5
3  71        10
3  72         7
3  73         9
3  74         4
3  75         8
3  76         6
3  77         7
3  78         6
3  79         7
3  80         8
3  81         6
3  82        14
3  83         8
3  84        14
3  85        30
3  86        11
3  87        19
3  88         8
3  89         6
3  90        12
3  91         4
3  92         3
3  93        10
3  94      10.5
3  95        16
3  96         4
3  97         4
3  98        10
3  99         6
3 100       5.5
3 101         7
3 102         4
3 103         6
3 104        11
3 105        13
3 106        20
3 107        10
3 108         2
3 109        10
3 110        20
3 111         8
3 112        16
3 113        10
3 114         3
3 115        15
3 116        10
3 117         6
3 118         4
3 119         4
3 120         8
3 121        11
3 122         9
3 123        19
3 124         3
3 125        14
3 126        14
3 127         6
3 128         5
3 129        22
3 130         5
3 131         7
3 132         4
3 133         7
3 134         9
3 135         5
3 136        14
3 137         6
3 138         7
3 139        16
3 140        10
3 141        11
3 142         7
3 143      28.5
3 144        10
3 145         8
3 146        16
3 147        12
3 148        10
3 149        12
3 150         6
3 151        11
3 152       4.5
3 153         7
3 154        17
3 155        13
3 156         3
3 157         5
3 158        15
3 159        30
3 160         9
3 161        14
3 162        12
3 163         5
3 164         8
3 165        10
3 166         5
3 167         5
3 168         6
3 169        11
3 170         7
3 171        10
3 172        14
3 173         8
3 174        22
3 175         5
3 176         8
3 177         7
3 178        10
3 179         5
3 180        11
3 181         4
3 182        28
3 183         9
3 184        15
3 185        10
3 186         6
3 187         5
3 188         8
3 189         9
3 190        15
3 191        12
3 192         8
3 193         5
3 194        16
3 195         6
3 196        11
3 197         5
3 198        15
3 199         5
3 200         6
3 201         8
3 202        14
3 203         5
3 204        17
3 205         8
3 206        13
3 207        13
3 208         5
3 209         5
3 210         7
3 211         4
3 212         6
3 213         7
3 214         9
3 215        17
3 216         8
3 217        10
3 218         7
3 219         3
3 220       5.5
3 221         7
3 222         3
3 223         7
3 224         4
3 225        13
3 226         4
3 227        11
3 228         5
3 229         9
3 230         8
3 231         4
3 232        16
3 233        13
3 234         7
3 235        14
3 236         9
3 237         5
3 238         5
3 239        12
3 240         5
3 241         5
3 242         4
3 243         4
3 244        13
3 245         9
3 246         6
3 247        10
3 248         9
3 249         4
3 250         2
3 251         6
3 252         1

label values Interventionalist Interventionalist
label def Durchfuehrer 2 "Joe", modify
label def Durchfuehrer 3 "Karl", modify

[Nick Cox]

Thanks for the data example. Still seems wide open to me -- in terms of what you want and also for others to pitch in.

[Michael DeLong]

Finding the point where levelling of the slope of the curve occurs?

[Nick Cox]

Amount of dose is, I gather, an input. Nothing in #6 looks even approximately monotonic except the time counter.

[Michael DeLong]

Can I calculate new variables for each Interventionalist containing the mean AmountOfDose for blocks of time (e.g. 1-10, 11-20 and so on?)