Examination of variability induced by experimental equipment

It is possible that the heterogeneity between box 1 and 2, respectively, is due to randomness. In that case the variance within each box is expected to be chi-square distributed. It is also a possibility that a disparity in the variation of the different boxes is due to some atypical groups of animals and, thus, is purely fortuitous.

The following model is fitted the data for each box

ln(Y_ijk) = µ + p_i + β₁ · t_j + β_i2 · pt_ij + A_k + ε_ijk, where

i = 1,…,2, j = -130,…,105, k = 1,…,32, ),

, 0 (

~ _A²

k N

A σ εijk ~^N(0,σ_ε²),

µ, pi and tj are the fixed effects for the intercept, period and slope concerning time, respectively. Ak is the random effect of the kth animal and εijk is the within-group error.

The residual components are then depicted against the chi-square distribution (figure 4.7.a). Furthermore, for each animal the following model is fitted the data

ln(Y_ij) = µ + p_i + β₁· t_j + β_i2· pt_ij + ε_ij where

i = 1,…,2, j = -130,…,105, εij ~^N(0,σ_ε²),

µ, pi and tj are the fixed effects for the intercept, period and slope concerning time, respectively and εij is the within-group error. These residual components are similarly showed against the chi-square distribution (figure 4.7.b). It is seen that the largest variance is more than twice the size of the smallest variance which is a remarkable difference. In the first plot, box 2 is only slightly deviant from the remainder but box 1

seems to be at a higher level than expected. In the second plot it is seen that box 1 and 2 is almost consistently positioned at the top and bottom of the plot, respectively. This pattern indicates that the variance heterogeneity may be on account of the boxes per se and not because of atypical animals in the different groups.

Based on these findings it was decided to do an additional experiment in order to find out how the boxes affect the response. Alternatively to the usual rats an ‘artificial rat’ was used, which is a squared box with a spring inside that is triggered in response to an acoustic cue above a certain threshold. The movement of the artificial rat is assumed relatively constant for the different activations.

The experiment was designed with the attempt to resemble the fear conditioning experiment in order make this experiment as supportive as possible for the evaluation of fear conditioning and to optimise the comparability of the results from the two experiments. All boxes were calibrated and the artificial rat was placed in a box and a tone occurred whereupon the response was measured for one second. This constitutes one trial and it was replicated successively 40 times. This was done for each box and the measurements of all boxes constituted a round. It was repeated five times in order to examine the progression of the measurements for a time period resembling the approximate length of a standard experiment that have five rounds. This procedure was iterated four days with no calibration of the boxes meanwhile in order to access the

Figure 4.7. (a) Residual components from model ‘MB’ depicted in a chi-square QQ-plot (b) Residual components from model ‘MA’ in a chi-square QQ-plot. In both cases the residual components from box 1 and box 2 have the highest and lowest values, respectively.

development of the response values throughout the period of a fear conditioning experiment.

The raw data from the first trial of the first round and box is shown in figure 4.8.

The waves of the spring are detectable but the sampling frequency is inadequate in order to make a proper description of the response curve. In order to compare the different boxes it is convenient to calculate a summary measure from each trial but as seen from the graph neither the maximum nor the average value seems to be completely reliable since the response values are sensitive to the time of measurement. Still, some tendencies might be possible to see from the experiment, although, ideally a new experiment should be carried out with a sampling interval at e.g. 1 millisecond.

The average value was calculated for the first 400 ms of each trial. The 40 average values from each round and each day were depicted vs. the ln-transformed response values with different colours assigned to each box in figure 4.9. Some outliers are seen but in the cases where the response is at a constantly low level throughout a round it is probably due to a loose connection to the artificial rat. For the individual low response points it may be due to a measurement error, but it might also be caused by noise in the laboratory that triggered the spring and due to a lag-phase rendered it

Figure 4.8. The response of the artificial rat from the system test. The wave is subsided after

insensitive to the acoustic cue furnished by the box. Because of this uncertainty the low values are removed in the future calculations.

At day 1 the response values of system 1 consistently drift downwards whereas the opposite is the case for system 2. Moreover, it is seen that some of the boxes are kept at a fairly stabile level, e.g. box 6 and 8, whereas box 1, 4 and 7, have a more fluctuating response pattern. Interestingly, over the days the variability of the response is decreasing and the smallest deviation is seen at day 4.

In order to take account for the shifting values it may be advantageous to include the interaction between box and round in a model. However, this is exactly what is done when ‘animal’ is included and, thus, ‘animal’ and the ‘box’-‘round’ interaction will be completely confounded. This imply that the estimated variance component for ‘animal’

Figure 4.9.a The response from the system test. A considerable variability is seen for the different rounds.

In general, the boxes from system 1 yields lower values than the boxes from system 2

contains some variability caused by the heterogeneity of the boxes in different rounds and it is not possible with the current design of fear conditioning to separate the effect of these two terms.

In order to compare variation for each box throughout a day the model

ln(Y_i) = µ + ε_i, i = 1,…,200, εi ~ ^N(0,σ_ε²),

is fitted each box and day and the residual components are depicted against the chi-square distribution (figure 4.10). It is seen that no box has a variation at a consistently higher or lower level than the others and this also holds true for box 1 and 2 that were different from the other boxes in the fear conditioning α5IA-II experiment. Because of

Figure 4.9.b. The response pattern for the system test for day 3 and 4. The response pattern is stabilized slightly at the end of the experiment.

the inconsistency in the graphs as well as the uncertainty in the response values it is decided not to take any further action regarding the individual boxes at present. Still, at some point it could be interesting to redo the experiment with a more appropriate setting in order to evaluate the experimental equipment.

Regarding the system effect it apparently induces a strong bias and this factor will therefore be included in a model and tested for statistical significance.