Minitab Project Report

Hana's click rates 10/28/10

This first test used the raw data.  If you look at the diagnostic plots you will see that it does not meet the assumptions of the ANOVA

General Linear Model: Controlled rate versus Behavior

Factor     Type Levels Values 
Behavior  fixed      5 Foraging    Milling     Resting     Socializing
                       Traveling  

Analysis of Variance for Controll, using Adjusted SS for Tests

Source     DF     Seq SS     Adj SS     Adj MS       F      P
Behavior    4     149240     149240      37310    3.08  0.018
Error     149    1802234    1802234      12096
Total     153    1951474  

Unusual Observations for Controll

Obs  Controll       Fit      SE Fit  Residual   St Resid
 34   327.600    79.829      15.251   247.771      2.27R 
 98   117.250   154.850      49.184   -37.600     -0.38 X
 99    32.250   154.850      49.184  -122.600     -1.25 X
100    50.250   154.850      49.184  -104.600     -1.06 X
101   439.500   154.850      49.184   284.650      2.89RX
102   135.000   154.850      49.184   -19.850     -0.20 X
104   721.500   118.588      15.251   602.912      5.54R 
107   497.500   118.588      15.251   378.912      3.48R 
109   422.000   118.588      15.251   303.412      2.79R 
134   653.500   118.588      15.251   534.912      4.91R 

R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.

Tukey 95.0% Simultaneous Confidence Intervals
Response Variable Controll
All Pairwise Comparisons among Levels of Behavior                              

Behavior = Foraging subtracted from:

Behavior     Lower    Center     Upper  -----+---------+---------+---------+-
Milling     -109.1    -31.74     45.60        (-----*------) 
Resting     -109.9    -33.79     42.36        (-----*------) 
Socializ     -67.4     75.02    217.39           (-----------*-----------) 
Travelin     -20.9     38.76     98.39               (----*----) 
                                        -----+---------+---------+---------+-
                                           -120         0       120       240

Behavior = Milling subtracted from:

Behavior     Lower    Center     Upper  -----+---------+---------+---------+-
Resting     -92.73    -2.051     88.63         (-------*------) 
Socializ    -43.89   106.759    257.41             (------------*-----------) 
Travelin     -6.84    70.497    147.83                (------*-----) 
                                        -----+---------+---------+---------+-
                                           -120         0       120       240

Behavior = Resting subtracted from:

Behavior     Lower    Center     Upper  -----+---------+---------+---------+-
Socializ    -41.23    108.81     258.8              (-----------*------------) 
Travelin     -3.60     72.55     148.7                 (-----*-----) 
                                        -----+---------+---------+---------+-
                                           -120         0       120       240

Behavior = Socializ subtracted from:

Behavior     Lower    Center     Upper  -----+---------+---------+---------+-
Travelin    -178.6    -36.26     106.1  (-----------*-----------) 
                                        -----+---------+---------+---------+-
                                           -120         0       120       240

Tukey Simultaneous Tests
Response Variable Controll
All Pairwise Comparisons among Levels of Behavior                              

Behavior = Foraging subtracted from:

Level        Difference       SE of             Adjusted
Behavior       of Means  Difference   T-Value    P-Value
Milling          -31.74       27.97    -1.135     0.7879
Resting          -33.79       27.54    -1.227     0.7357
Socializ          75.02       51.49     1.457     0.5920
Travelin          38.76       21.57     1.797     0.3792

Behavior = Milling subtracted from:

Level        Difference       SE of             Adjusted
Behavior       of Means  Difference   T-Value    P-Value
Resting          -2.051       32.80  -0.06254     1.0000
Socializ        106.759       54.49   1.95933     0.2911
Travelin         70.497       27.97   2.52033     0.0916

Behavior = Resting subtracted from:

Level        Difference       SE of             Adjusted
Behavior       of Means  Difference   T-Value    P-Value
Socializ         108.81       54.27     2.005     0.2687
Travelin          72.55       27.54     2.634     0.0694

Behavior = Socializ subtracted from:

Level        Difference       SE of             Adjusted
Behavior       of Means  Difference   T-Value    P-Value
Travelin         -36.26       51.49   -0.7042     0.9553

Residuals vs Fits for Controll

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We therefore square root transformed the data, but the data still did not meet the assumptions very well (see plots below).

General Linear Model: sqrt(rate) versus Behavior

Factor     Type Levels Values 
Behavior  fixed      5 Foraging    Milling     Resting     Socializing
                       Traveling  

Analysis of Variance for sqrt(rat, using Adjusted SS for Tests

Source     DF     Seq SS     Adj SS     Adj MS       F      P
Behavior    4     342.35     342.35      85.59    3.12  0.017
Error     149    4084.85    4084.85      27.42
Total     153    4427.20  

Unusual Observations for sqrt(rat

Obs  sqrt(rat       Fit      SE Fit  Residual   St Resid
 98   10.8282   11.2358      2.3416   -0.4076     -0.09 X
 99    5.6789   11.2358      2.3416   -5.5569     -1.19 X
100    7.0887   11.2358      2.3416   -4.1471     -0.89 X
101   20.9643   11.2358      2.3416    9.7284      2.08RX
102   11.6190   11.2358      2.3416    0.3831      0.08 X
104   26.8608    8.1779      0.7261   18.6828      3.60R 
107   22.3047    8.1779      0.7261   14.1268      2.72R 
109   20.5426    8.1779      0.7261   12.3647      2.38R 
134   25.5636    8.1779      0.7261   17.3857      3.35R 

R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.

Tukey 95.0% Simultaneous Confidence Intervals
Response Variable sqrt(rat
All Pairwise Comparisons among Levels of Behavior                              

Behavior = Foraging subtracted from:

Behavior     Lower    Center     Upper  -----+---------+---------+---------+-
Milling     -5.889    -2.207   1.47480         (----*----) 
Resting     -7.224    -3.599   0.02621       (----*----) 
Socializ    -3.922     2.856   9.63437           (---------*---------) 
Travelin    -3.041    -0.202   2.63741             (---*---) 
                                        -----+---------+---------+---------+-
                                           -7.0       0.0       7.0      14.0

Behavior = Milling subtracted from:

Behavior     Lower    Center     Upper  -----+---------+---------+---------+-
Resting     -5.709    -1.392     2.925         (-----*-----) 
Socializ    -2.109     5.063    12.235              (---------*---------) 
Travelin    -1.676     2.005     5.687               (----*----) 
                                        -----+---------+---------+---------+-
                                           -7.0       0.0       7.0      14.0

Behavior = Resting subtracted from:

Behavior     Lower    Center     Upper  -----+---------+---------+---------+-
Socializ   -0.6879     6.455    13.598                (---------*---------) 
Travelin   -0.2278     3.397     7.022                 (----*----) 
                                        -----+---------+---------+---------+-
                                           -7.0       0.0       7.0      14.0

Behavior = Socializ subtracted from:

Behavior     Lower    Center     Upper  -----+---------+---------+---------+-
Travelin    -9.836    -3.058     3.720   (---------*--------) 
                                        -----+---------+---------+---------+-
                                           -7.0       0.0       7.0      14.0

Tukey Simultaneous Tests
Response Variable sqrt(rat
All Pairwise Comparisons among Levels of Behavior                              

Behavior = Foraging subtracted from:

Level        Difference       SE of             Adjusted
Behavior       of Means  Difference   T-Value    P-Value
Milling          -2.207       1.332    -1.657     0.4636
Resting          -3.599       1.311    -2.745     0.0523
Socializ          2.856       2.452     1.165     0.7712
Travelin         -0.202       1.027    -0.196     0.9997

Behavior = Milling subtracted from:

Level        Difference       SE of             Adjusted
Behavior       of Means  Difference   T-Value    P-Value
Resting          -1.392       1.561   -0.8914     0.8997
Socializ          5.063       2.594    1.9519     0.2949
Travelin          2.005       1.332    1.5059     0.5603

Behavior = Resting subtracted from:

Level        Difference       SE of             Adjusted
Behavior       of Means  Difference   T-Value    P-Value
Socializ          6.455       2.584     2.499     0.0965
Travelin          3.397       1.311     2.591     0.0772

Behavior = Socializ subtracted from:

Level        Difference       SE of             Adjusted
Behavior       of Means  Difference   T-Value    P-Value
Travelin         -3.058       2.452    -1.247     0.7236

Residuals vs Fits for sqrt(rat

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We log tranformed the data,  but that was too much of a tranform so Val suggested using a double square root transform with the following results.  The data now seem to meet the assumptions of the ANOVA

General Linear Model: sqrt(sqrt(rate)) versus Behavior

Factor     Type Levels Values 
Behavior  fixed      5 Foraging    Milling     Resting     Socializing
                       Traveling  

Analysis of Variance for sqrt(sqr, using Adjusted SS for Tests

Source     DF     Seq SS     Adj SS     Adj MS       F      P
Behavior    4     25.375     25.375      6.344    5.05  0.001
Error     149    187.155    187.155      1.256
Total     153    212.530  

Unusual Observations for sqrt(sqr

Obs  sqrt(sqr       Fit      SE Fit  Residual   St Resid
 72   0.00000   2.30322     0.23894  -2.30322     -2.10R 
 73   0.00000   2.30322     0.23894  -2.30322     -2.10R 
 98   3.29062   3.26469     0.50121   0.02593      0.03 X
 99   2.38305   3.26469     0.50121  -0.88165     -0.88 X
100   2.66247   3.26469     0.50121  -0.60223     -0.60 X
101   4.57867   3.26469     0.50121   1.31398      1.31 X
102   3.40866   3.26469     0.50121   0.14396      0.14 X
104   5.18274   2.47867     0.15542   2.70407      2.44R 
107   4.72279   2.47867     0.15542   2.24412      2.02R 
111   0.00000   2.47867     0.15542  -2.47867     -2.23R 
130   0.00000   2.47867     0.15542  -2.47867     -2.23R 
132   0.00000   2.47867     0.15542  -2.47867     -2.23R 
134   5.05605   2.47867     0.15542   2.57738      2.32R 

R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.

Tukey 95.0% Simultaneous Confidence Intervals
Response Variable sqrt(sqr
All Pairwise Comparisons among Levels of Behavior                              

Behavior = Foraging subtracted from:

Behavior     Lower    Center     Upper  -----+---------+---------+---------+-
Milling     -1.331    -0.543    0.2452        (----*-----) 
Resting     -1.940    -1.164   -0.3880    (----*----) 
Socializ    -1.032     0.419    1.8695          (---------*--------) 
Travelin    -0.975    -0.367    0.2403          (----*---) 
                                        -----+---------+---------+---------+-
                                           -1.5       0.0       1.5       3.0

Behavior = Milling subtracted from:

Behavior     Lower    Center     Upper  -----+---------+---------+---------+-
Resting     -1.545   -0.6211    0.3030       (-----*-----) 
Socializ    -0.574    0.9615    2.4966             (---------*----------) 
Travelin    -0.613    0.1754    0.9635             (----*----) 
                                        -----+---------+---------+---------+-
                                           -1.5       0.0       1.5       3.0

Behavior = Resting subtracted from:

Behavior     Lower    Center     Upper  -----+---------+---------+---------+-
Socializ   0.05361    1.5826     3.112                 (----------*---------) 
Travelin   0.02060    0.7966     1.573                 (----*----) 
                                        -----+---------+---------+---------+-
                                           -1.5       0.0       1.5       3.0

Behavior = Socializ subtracted from:

Behavior     Lower    Center     Upper  -----+---------+---------+---------+-
Travelin    -2.237   -0.7860    0.6648  (---------*--------) 
                                        -----+---------+---------+---------+-
                                           -1.5       0.0       1.5       3.0

Tukey Simultaneous Tests
Response Variable sqrt(sqr
All Pairwise Comparisons among Levels of Behavior                              

Behavior = Foraging subtracted from:

Level        Difference       SE of             Adjusted
Behavior       of Means  Difference   T-Value    P-Value
Milling          -0.543      0.2850    -1.904     0.3195
Resting          -1.164      0.2807    -4.147     0.0005
Socializ          0.419      0.5248     0.798     0.9309
Travelin         -0.367      0.2198    -1.672     0.4546

Behavior = Milling subtracted from:

Level        Difference       SE of             Adjusted
Behavior       of Means  Difference   T-Value    P-Value
Resting         -0.6211      0.3342    -1.858     0.3445
Socializ         0.9615      0.5553     1.732     0.4179
Travelin         0.1754      0.2850     0.616     0.9724

Behavior = Resting subtracted from:

Level        Difference       SE of             Adjusted
Behavior       of Means  Difference   T-Value    P-Value
Socializ         1.5826      0.5530     2.862     0.0382
Travelin         0.7966      0.2807     2.838     0.0407

Behavior = Socializ subtracted from:

Level        Difference       SE of             Adjusted
Behavior       of Means  Difference   T-Value    P-Value
Travelin        -0.7860      0.5248    -1.498     0.5655

Residuals vs Fits for sqrt(sqr

Executing from file: C:\Program Files\MTBWIN\MACROS\GFMAIN.MAC
 
Macro is running ... please wait

Just for fun we ran the K-W test on the raw data and also found significance.

Kruskal-Wallis Test: Controlled rate versus Behavior

Kruskal-Wallis Test on Controll

Behavior    N    Median    Ave Rank         Z
Foraging   52    56.667        87.3      1.95
Milling    22    41.167        66.3     -1.27
Resting    23     2.000        56.0     -2.50
Socializ    5   117.250       106.2      1.46
Travelin   52    59.750        79.2      0.33
Overall   154                  77.5

H = 11.36  DF = 4  P = 0.023
H = 11.37  DF = 4  P = 0.023 (adjusted for ties)

We then ran an ANCOVA with ambient noise as a covariate on all the data to see what kind of effect ambient noise might have.  These are the results.

General Linear Model: sqrt(sqrt(rate)) versus Behavior

Factor     Type Levels Values 
Behavior  fixed      5 Foraging    Milling     Resting     Socializing
                       Traveling  

Analysis of Variance for sqrt(sqr, using Adjusted SS for Tests

Source     DF     Seq SS     Adj SS     Adj MS       F      P
Ambient     1     18.721     24.034     24.034   21.81  0.000
Behavior    4     30.688     30.688      7.672    6.96  0.000
Error     148    163.121    163.121      1.102
Total     153    212.530  

Term             Coef   SE Coef        T      P
Constant       -2.827     1.150    -2.46  0.015
Ambient       0.04889   0.01047     4.67  0.000

Unusual Observations for sqrt(sqr

Obs  sqrt(sqr       Fit      SE Fit  Residual   St Resid
 98   3.29062   3.34487     0.46982  -0.05424     -0.06 X
 99   2.38305   3.29598     0.46955  -0.91293     -0.97 X
100   2.66247   3.28131     0.46952  -0.61885     -0.66 X
101   4.57867   3.22754     0.46957   1.35113      1.44 X
102   3.40866   3.17377     0.46991   0.23489      0.25 X
104   5.18274   2.38371     0.14700   2.79902      2.69R 
107   4.72279   2.44727     0.14574   2.27552      2.19R 
111   0.00000   2.58904     0.14749  -2.58904     -2.49R 
130   0.00000   2.73569     0.15564  -2.73569     -2.63R 
132   0.00000   2.89213     0.17040  -2.89213     -2.79R 
134   5.05605   2.46193     0.14563   2.59412      2.50R 

R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.

Tukey 95.0% Simultaneous Confidence Intervals
Response Variable sqrt(sqr
All Pairwise Comparisons among Levels of Behavior                              

Behavior = Foraging subtracted from:

Behavior     Lower    Center     Upper  -----+---------+---------+---------+-
Milling     -1.668    -0.900   -0.1321      (----*----) 
Resting     -1.768    -1.037   -0.3067     (----*----) 
Socializ    -1.580    -0.176    1.2276      (---------*--------) 
Travelin    -1.689    -1.006   -0.3223      (---*----) 
                                        -----+---------+---------+---------+-
                                           -1.5       0.0       1.5       3.0

Behavior = Milling subtracted from:

Behavior     Lower    Center     Upper  -----+---------+---------+---------+-
Resting     -1.049   -0.1374    0.7744          (-----*-----) 
Socializ    -0.721    0.7237    2.1686            (---------*--------) 
Travelin    -0.862   -0.1055    0.6512           (----*----) 
                                        -----+---------+---------+---------+-
                                           -1.5       0.0       1.5       3.0

Behavior = Resting subtracted from:

Behavior     Lower    Center     Upper  -----+---------+---------+---------+-
Socializ   -0.6336   0.86103    2.3556             (---------*---------) 
Travelin   -0.8245   0.03187    0.8882            (----*-----) 
                                        -----+---------+---------+---------+-
                                           -1.5       0.0       1.5       3.0

Behavior = Socializ subtracted from:

Behavior     Lower    Center     Upper  -----+---------+---------+---------+-
Travelin    -2.188   -0.8292    0.5301  (--------*---------) 
                                        -----+---------+---------+---------+-
                                           -1.5       0.0       1.5       3.0

Tukey Simultaneous Tests
Response Variable sqrt(sqr
All Pairwise Comparisons among Levels of Behavior                              

Behavior = Foraging subtracted from:

Level        Difference       SE of             Adjusted
Behavior       of Means  Difference   T-Value    P-Value
Milling          -0.900      0.2778    -3.241     0.0126
Resting          -1.037      0.2643    -3.925     0.0012
Socializ         -0.176      0.5078    -0.347     0.9968
Travelin         -1.006      0.2471    -4.069     0.0007

Behavior = Milling subtracted from:

Level        Difference       SE of             Adjusted
Behavior       of Means  Difference   T-Value    P-Value
Resting         -0.1374      0.3298   -0.4165     0.9936
Socializ         0.7237      0.5226    1.3847     0.6384
Travelin        -0.1055      0.2737   -0.3854     0.9953

Behavior = Resting subtracted from:

Level        Difference       SE of             Adjusted
Behavior       of Means  Difference   T-Value    P-Value
Socializ        0.86103      0.5406    1.5928     0.5044
Travelin        0.03187      0.3097    0.1029     1.0000

Behavior = Socializ subtracted from:

Level        Difference       SE of             Adjusted
Behavior       of Means  Difference   T-Value    P-Value
Travelin        -0.8292      0.4916    -1.687     0.4454

We then ran an ANCOVA without the data from the 16th since that was perhaps driving the relationship due to it having quiet noise and only foraging.

General Linear Model: sqrt(sqrt(rate-no 16)) versus behavior (no 16th)

Factor     Type Levels Values 
behavior  fixed      5 Foraging    Milling     Resting     Socializing
                       Traveling  

Analysis of Variance for sqrt(sqr, using Adjusted SS for Tests

Source     DF     Seq SS     Adj SS     Adj MS       F      P
ambient     1     54.738     41.304     41.304   34.36  0.000
behavior    4      5.648      5.648      1.412    1.17  0.326
Error     110    132.214    132.214      1.202
Total     115    192.600  

Term             Coef   SE Coef        T      P
Constant       -7.102     1.647    -4.31  0.000
ambient       0.08572   0.01462     5.86  0.000

Unusual Observations for sqrt(sqr

Obs  sqrt(sqr       Fit      SE Fit  Residual   St Resid
 60   3.29062   3.40527     0.49088  -0.11465     -0.12 X
 61   2.38305   3.31955     0.49038  -0.93651     -0.96 X
 62   2.66247   3.29384     0.49032  -0.63137     -0.64 X
 63   4.57867   3.19955     0.49042   1.37913      1.41 X
 64   3.40866   3.10526     0.49105   0.30340      0.31 X
 66   5.18274   2.31218     0.15466   2.87056      2.64R 
 69   4.72279   2.42361     0.15232   2.29918      2.12R 
 73   0.00000   2.67219     0.15558  -2.67219     -2.46R 
 92   0.00000   2.92934     0.17037  -2.92934     -2.70R 
 94   0.00000   3.20364     0.19598  -3.20364     -2.97R 
 95   0.66874   2.86077     0.16542  -2.19203     -2.02R 
 96   5.05605   2.44932     0.15212   2.60673      2.40R 

R denotes an observation with a large standardized residual.
X denotes an observation whose X value gives it large influence.

Tukey 95.0% Simultaneous Confidence Intervals
Response Variable sqrt(sqr
All Pairwise Comparisons among Levels of behavior                              

behavior = Foraging subtracted from:

behavior     Lower    Center     Upper  ------+---------+---------+---------+
Milling     -1.003    0.0794    1.1622           (-------*------) 
Resting     -0.942    0.3065    1.5554            (-------*-------) 
Socializ    -0.963    0.6239    2.2108            (---------*----------) 
Travelin    -1.156   -0.2377    0.6801          (-----*------) 
                                        ------+---------+---------+---------+
                                            -1.5       0.0       1.5       3.0

behavior = Milling subtracted from:

behavior     Lower    Center     Upper  ------+---------+---------+---------+
Resting     -0.764    0.2271    1.2181             (------*-----) 
Socializ    -0.974    0.5445    2.0629            (---------*---------) 
Travelin    -1.124   -0.3172    0.4901           (----*----) 
                                        ------+---------+---------+---------+
                                            -1.5       0.0       1.5       3.0

behavior = Resting subtracted from:

behavior     Lower    Center     Upper  ------+---------+---------+---------+
Socializ    -1.297    0.3174    1.9318         (----------*----------) 
Travelin    -1.535   -0.5443    0.4462        (-----*------) 
                                        ------+---------+---------+---------+
                                            -1.5       0.0       1.5       3.0

behavior = Socializ subtracted from:

behavior     Lower    Center     Upper  ------+---------+---------+---------+
Travelin    -2.285   -0.8617    0.5617   (--------*---------) 
                                        ------+---------+---------+---------+
                                            -1.5       0.0       1.5       3.0

Tukey Simultaneous Tests
Response Variable sqrt(sqr
All Pairwise Comparisons among Levels of behavior                              

behavior = Foraging subtracted from:

Level        Difference       SE of             Adjusted
behavior       of Means  Difference   T-Value    P-Value
Milling          0.0794      0.3906    0.2034     0.9996
Resting          0.3065      0.4506    0.6804     0.9603
Socializ         0.6239      0.5725    1.0899     0.8114
Travelin        -0.2377      0.3311   -0.7179     0.9520

behavior = Milling subtracted from:

Level        Difference       SE of             Adjusted
behavior       of Means  Difference   T-Value    P-Value
Resting          0.2271      0.3575     0.635     0.9690
Socializ         0.5445      0.5478     0.994     0.8576
Travelin        -0.3172      0.2912    -1.089     0.8118

behavior = Resting subtracted from:

Level        Difference       SE of             Adjusted
behavior       of Means  Difference   T-Value    P-Value
Socializ         0.3174      0.5824     0.545     0.9823
Travelin        -0.5443      0.3573    -1.523     0.5498

behavior = Socializ subtracted from:

Level        Difference       SE of             Adjusted
behavior       of Means  Difference   T-Value    P-Value
Travelin        -0.8617      0.5135    -1.678     0.4517

Residuals vs Fits for sqrt(sqr

Executing from file: C:\Program Files\MTBWIN\MACROS\GFMAIN.MAC
 
Macro is running ... please wait