### Article

## Statistical Indicator for the Evaluation of Human Gait Analysis by Ground Reaction Forces

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Published: | September 8, 2005 |
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### Outline

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#### Introduction and purpose

In addition to the subjective clinical evaluation of human gait made by a physician an objective approach that can reveal processes hidden to the human eye is desired. This work characterises the gait of pathological subjects quantitatively in respect to subjects whose gait is considered to be normal. These subjects had different parts of the foot surgically reconstructed. A statistical method is applied to the recorded data which enables insight into the different structures.

#### Methods and Materials

The ground reaction forces were measured for a control group (27 subjects with 3 trials each) and pathologic subjects (4 operated at forefoot, 10 at rear foot, 6 at dorsum). The parameters of interest (as described in [Ref. 1]) were extracted from the measured trajectories: The force parameters (F1-F6) and the corresponding time instants (T1-T6).

To the data of the subjects of the control group a principal component analysis was applied, which is a linear transformation that maximises the variance for the variables in the new space, called principal components (PCs). With this method redundancy is reduced and sometimes underlying mechanisms can be revealed. 2/3 of the control group’s subjects were used for modelling, 1/3 was later used for cross validation. Initially Hotelling’s T-test was applied to clear the set from outliers, because these would have distorted the model. To point out the structure of the PCs they were noted after a simplification method mentioned in [Ref. 2], where strong positive loadings are marked with a “+”, strong negative ones with a “-“. This way a meaning could be given to them. E.g., the first PC contrasts the corresponding vertical and fore-after forces at loading response (F1 and F4) and at terminal stance (F3 and F6) with the forces at mid stance (F2 and F5). I.e. when the values of F1, F4, F3 and F6 are high, the values of F2 and F5 are low and vice versa.

To get a first impression of the differences between the groups, PCs were calculated also for the pathologic groups. Indeed, their structure differs to some extent very much from the control group. The PCs’ composition of the rear foot and the forefoot group is simpler; the first component represents the time instants T3 and T6. Only the dorsum group shows a strong similarity to the control group.

For the PC model a certain number of the 12 PCs of the control group had to be selected. Trying out different combinations, the first 11 components turned out to be suitable to represent the control group and to characterise the pathologic subjects. This selection was cross validated, i.e. 1/3 of the control group that had not been used for the calculation of the model, were projected into the PC space with 11 components and back into the original space, and the differences were calculated between this data and the original samples, defined as the residuals.

In order to assess the samples in the PC space the difference measures T^{2} and Q were applied [Ref. 3]. Thereby T^{2} is the Mahalanobis distance between a sample and the centre of the PC space, demonstrating the accordance with the model. I.e. the higher the T^{2} value, the higher the probability of the sample not belonging to the population of the training set. Q is the residual, the perpendicular distance between the sample and the hyper-plane represented by the chosen PCs.

In this T^{2}-Q plane, the samples of the control group and the pathologic groups can be represented. In order to find out how the parameters influence the position of a sample, test samples were created with mean values in all but one parameter. Interestingly some parameters had almost no influence, whereas high/low values in the force parameter F3 and especially in the time parameters T1, T2, T3 and T6 resulted in rather extreme values in T^{2} and Q, and with that in a position in the T^{2}-Q plane far away from the samples of the control group.

The application of the model on the control group and the pathologic groups revealed that most of the rear foot objects are distinguished from “normal” by their T^{2} value; the Q values are similar to those of the control group. Many forefoot operated subjects differ in both T^{2} and Q values. The dorsum operated subjects have “normal” Q values, some differ in T^{2}. This is the group of which most samples are close to the control group.

Some samples were used to verify that the position in the T^{2}-Q plane can give information about the kind of deviation from the control group. In order to obtain a numerical description that summarizes the consistency of the sample with the “normal gait”, the Mahalanobis distance between a sample and the training set in the T^{2}-Q plane is used. With this statistical indicator a region for “normal gait” can be defined. No conclusions about the distributions regarding the T^{2} and Q values could be drawn, because it showed that not all of the 12 original variables are approximately Gaussian distributed. Thus a nonparametric method, the bootstrap, was applied (with 2000 resamples using all subjects from the control group) to determine a 95% confidence interval for the 95% quantile, i.e. the region containing 95% of the samples of the control group. The plot of the borders together with the samples of the forefoot operated subjects can be seen in figure 1 [Fig. 1].

#### Results

Most of the subjects from the rear foot and the forefoot group are distant in the model from the control group, whereas samples from the dorsum group are rather near. The parameter values of the dorsum samples and the PCs of the dorsum group show that it has a variance structure which is more similar to that of the control group than that of the rear foot and the forefoot group. This confirms that subjects operated at the dorsum have a gait that is rather similar to “normal” gait, whereas injuries at the plantar area have a severe influence on the gait.

The T^{2} and Q values give a useful indication about which parameters characterise “unnormal” gait: the values of some dynamic parameters have a great influence on the position of a trial in the T^{2}−Q plane while others are almost irrelevant. Namely the force variable F3 and the time variables T1, T2, T3 and T6 turned out to be important. Indeed in these parameters the dorsum group is similar to the control group.

#### Discussion and Conclusion

The principal component analysis is an appropriate method of finding out which variables (F3, T1, T2, T3 and T6) of the recorded data are important in a sense of detecting “unnormality”. Therefore the Mahalanobis distance in the T^{2}−Q plane can be used as a statistical indicator to characterise the gait of different subjects in respect to their accordance with the model.

#### Thanks to

F. Verdini, Dipartimento di Ingegneria Informatica Gestionale e dell’Automazione, Università Politecnica delle Marche, Ancona, Italy

### References

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- Chao A, Laughman R K, Schneider E, Stauffer R N. Normative data of knee joint motion and ground reaction forces in adult level walking. Biomechanics, 16(3) 219-233, 1983
- 2.
- Jolliffe J E. A User's Guide To Principal Components. Springer Verlag, 1986
- 3.
- Deluzio K J, Wyss U P, Zee B, Costigan P A, Sorbie C. Principal component models of knee kinematics and kinetics: Normal vs. pathologic gait patterns. Human Movement Science, 16(2-3): 201-217, April 1997