21

Japanese Journal of Comprehensive Rehabilitation Science (2017)

Original Article

The stratification of motor FIM and cognitive FIM and the creationof four prediction formulas to enable higher prediction accuracy ofmultiple linear regression analysis with motor FIM gain as theobjective variable ― An analysis of the Japan Rehabilitation Database
Makoto Tokunaga, MD, PhD,1 Kenichi Tori, RPT,1 Hiroshi Eguchi, RPT,1 Youko Kado, RPT,1
Yuki Ikejima, RPT,1 Miyuki Ushijima, RPT,1 Shinko Miyabe, OTR,1 Shinya Tsujimoto, RPT,1
Emiko Fukuda, OTR1
Department of Sogo Rehabilitation, Kumamoto Kinoh Hospital, Kumamoto, Japan

1

ABSTRACT
Tokunaga M, Tori K, Eguchi H, Kado Y, Ikejima Y,
Ushijima M, Miyabe S, Tsujimoto S, Fukuda E. The
stratification of motor FIM and cognitive FIM and thecreation of four prediction formulas to enable higher
prediction accuracy of multiple linear regression
analysis with motor FIM gain as the objective variable― An analysis of the Japan Rehabilitation Database.
Jpn J Compr Rehabil Sci 2017; 8: 21-29.
Objective: The aim of our study was to stratify thecontributing factors in order to increase the prediction
accuracy of the multiple linear regression analysis
with motor FIM gain as the objective variable.
Methods: The subjects for our study were 2,542
stroke patients. In the multiple linear regression
analysis with motor FIM gain as the objective variable,
eight contributing factors were stratified. Prediction
formulas were created and the correlation between themeasured motor FIM gain values and the predictedvalues was investigated.
Results: The correlation coefficient was higher withthe stratification of gender (0.509), stroke type (0.512),
number of hospital days (0.516), days from onset toadmission (0.518), modified Rankin Scale before
onset (0.520), age (0.541), cognitive FIM at admission(0.588) and motor FIM at admission (0.641), than withthe use of one prediction formula (0.507), and it was
0.653 with stratification into four groups with the two
factors of motor FIM and cognitive FIM at admission.
Correspondence: Makoto Tokunaga, MD, PhD
Department of Rehabilitation, Kumamoto Kinoh Hospital,
6-8-1 Yamamuro, Kita-ku, Kumamoto 860-8518, Japan.
E-mail: tokunaga@juryo.or.jp
Accepted: February 8, 2017.
No benefits in any form have been or will be received fromany commercial party related directly or indirectly to thesubject of this manuscript.
©Kaifukuki Rehabilitation Ward Association 2017
doi.org/10.11336/jjcrs.8.21

Conclusion: By stratifying the contributing factors,
we were able to increase the prediction accuracy of
motor FIM gain.
Key words: Functional Independence Measure, FIM
gain, multiple linear regression analysis, stratification,
strokeIntroduction
As the objective variable of multiple linear
regression analysis for the prediction of functional
outcome, the Functional Independence Measure (FIM)
score at discharge and FIM gain (FIM score atdischarge－FIM score at admission) are used. Meyer
et al. [1] conducted reviews of 27 reports, whichcontained 63 formulas predicting the functional
outcome of acute stroke patients using multiple linear
regression analysis. The objective variables were: FIM
score at discharge in 33 prediction formulas, FIM gainin 20 formulas, FIM efficiency (FIM gain/number ofhospital days) in three formulas, Barthel Index (BI)score at discharge in five formulas and BI efficiency
in two formulas. It is considered that there werefewer reports using FIM gain than reports using FIM
score at discharge due to the effect of the size of thecoefficient of determination or R2, meaning to what
degree the explanatory variable could explain theobjective variable. In the case of predicting FIM scoreat discharge, the average R2 was 0.65 (min. 0.35– max.
0.82). In contrast, the average was small at 0.22 (0.08–0.4) in the case of predicting FIM gain [1].
For that reason, in a multiple linear regression
analysis predicting FIM gain, a technique forincreasing prediction accuracy is required.
One approach is the technique of ‘creating multiple
prediction formulas.’ In a multiple linear regression
analysis, the prediction formula Y = aX1 + bX2 + cX3(Y: objective variable, X1–X3: explanatory variable,
Jpn J Compr Rehabil Sci Vol 8, 2017

22 Tokunaga M et al.: Increasing the prediction accuracy of multiple linear regression analysis for the prediction of FIM gaina–c: partial regression coefficient) is obtained. This
formula assumes a direct relationship between thecontributory factor (explanatory variable) and theobjective variable. For a case in which there is no
direct relationship between the explanatory variable
and the objective variable, it is considered thatstratifying the factors and creating multiple prediction
formulas will increase the prediction accuracy. Thus,
in a multiple linear regression analysis predicting FIM
score at discharge, Inouye [2] stratified age into fivegroups, and created five prediction formulas. Sonoda
et al. [3] showed the need for stratifying FIM score atadmission into two groups (FIM score of 80 points orabove, and below 80 points), and Hirano et al. [4]stratified FIM at admission into three groups withthree prediction formulas. Tokunaga et al. [5] stratified
the three factors of motor FIM score at admission,
cognitive FIM score at admission and age, and createdeight prediction formulas. Regarding multiple linear
regression analysis predicting FIM gain, Wang et al.
[6] and Tokunaga et al. [7] stratified FIM score atadmission into two groups and created two prediction
formulas, and Imada et al. [8] stratified motor FIM
score and cognitive FIM at admission, creating three
prediction formulas. Nevertheless, in these reports [28] it was not clear to what degree the prediction
accuracy increased with multiple prediction formulas
as opposed to a single prediction formula.
Tokunaga et al. [9], in the prediction of FIM score atdischarge in stroke patients, showed that with thestratification of two groups according to age and withthree groups for FIM score at admission, for a total ofsix groups, and creating six prediction formulas,means that the correlation between the measured
values and predicted values of FIM score at discharge
is higher than with a single prediction formula (thecorrelation coefficient for multiple formulas was 0.893
and for a single formula 0.863). However, it isconsidered that the necessity of stratification of FIM
score at admission is greater with the prediction of
FIM gain than with the prediction of FIM score atdischarge. If a graph is created with FIM score atadmission on the horizontal axis and FIM score atdischarge on the vertical axis (while there is a slight
upward convex structure), the higher the FIM score atadmission, the higher the FIM score at discharge.
However, if FIM gain is placed on the vertical axis, amountain-shaped relationship is created (a non-direct
relationship between FIM score at admission and FIMgain) (Fig. 1). This is due to the fact that at the total
assistance level there are many patients for whom
improvement is difficult, and at the minimal assistancelevel the ceiling effect leads to smaller FIM gain, andcontrastingly, FIM gain is often large with moderateassistance patients [10]. For this reason, when
predicting FIM gain, it is considered necessary to
stratify FIM score at admission (even more so thanwhen predicting FIM score at discharge).
Jpn J Compr Rehabil Sci Vol 8, 2017

Figure 1. Relationship between motor FIM at
admission and motor FIM gain.
Data from 2,542 cases in this study.
Recently, Tokunaga et al. [11] showed that in
the prediction of motor FIM gain in stroke patients,
if a two-group stratification of motor FIM score atadmission and two prediction formulas are created,
there is a larger coefficient of correlation between themeasured and predicted motor FIM gain values incomparison to a single prediction formula with nostratification (correlation coefficients: 0.641 for theformer and 0.507 for the latter). However, in theprediction of motor FIM gain, it was not clear to what
degree the correlation between the measured andpredicted motor FIM gain values increased bystratifying many factors other than motor FIM score atadmission.
In this study, we used nationwide data on stroke
patients admitted to rehabilitation wards during therecovery period to predict motor FIM gain withmultiple linear regression analysis. Within thatprocess, we stratified age, gender, stroke type, modified
Rankin Scale (mRS) before onset, days from onset toadmission, motor FIM score at admission, cognitive
FIM score at admission and number of hospital days,
and created prediction formulas. The objective of thisstudy was to compare the correlation coefficient forthe measured and predicted motor FIM gain values.
Subjects and Methods
We used patient data from the Japan Rehabilitation
Database (JRD) created by the Japan Associationof Rehabilitation Database (JARD), which wasestablished in September 2012 with the goal of
constructing and using a rehabilitation database toTokunaga M et al.: Increasing the prediction accuracy of multiple linear regression analysis for the prediction of FIM
gain
23

Figure 2. Inclusion and exclusion criteria.
FIM, Functional Independence Measure; mRS, modified Rankin Scale; Numerical value, number of patients.
help improve rehabilitation medicine and care [12].
The groups that comprise JARD are: the Japanese
Association of Rehabilitation Medicine, the JapanesePhysical Therapy Association, the Japanese Associationof Occupational Therapists, and the Japanese
Association of Speech-Language-Hearing Therapists.
Data on patients who have suffered a stroke, hipfracture, or spinal cord injury are collected fromparticipating institutions throughout Japan [13]. In
JARD, all personal information is converted to data toprevent the identification of individuals. The present
study was conducted with the permission of JARD.
This epidemiological research was a retrospective
study. The subjects were selected from 6,322 stroke
patients registered with the JRD in April 2015 (stroke
patients hospitalized in Kaifukuki rehabilitation
wards). To reduce the influence of exceptional casesthat could be seen as outliers, the subjects were limited
to patients who fulfilled the following inclusion
criteria (Fig. 2): age 15 to 99 years, duration fromonset to hospital admission of 5 to 90 days, admitted to
Kaifukuki rehabilitation wards for 21 to 210 days,
motor FIM score at admission of 13 to 90 points(patients with 91 points excluded), FIM gain of 0 or
higher, and having entries for all items to be examined.
This screening obtained 2,542 subjects. The target
patients were the same patients as in our previous
report [11]. Table 1 shows their basic characteristics.



Table 1. Basic characteristics of the subjects.
Number of patients
Gender2,542
Male 1,492; female 1,050

Stroke type

Cerebral infarction 1,613
Cerebral hemorrhage 772
Subarachnoid hemorrhage 157

AgemRS before onset

69.3±12.9 (71)0.7±1.4 (0)Days from onset to hospital admission
Days in the hospital

36.3±15.2 (33)101.6±44.8 (100)Motor FIM score at admission
Cognitive FIM score at admission
Total FIM score at admission

45.7±23.2 (46)21.9±9.0 (23)67.6±29.9 (69)Motor FIM score at discharge
Cognitive FIM score at discharge
Total FIM score at discharge

65.7±23.1 (74)25.7±8.3 (28)91.4±29.9 (101)Motor FIM gain

20.0±14.8 (18)

Numerical value, Mean ± standard deviation or number of patients.

Investigation 1: Motor FIM gain according tostratified factors
We stratified the eight factors of age, gender, stroke
type, mRS before onset, days from onset to admission,
motor FIM score at admission, cognitive FIM score atadmission and number of hospital days. For stroke
type, there were three groups (cerebral infarction,
cerebral hemorrhage, subarachnoid hemorrhage);
however, for the other seven classifications, there were
two groups each. Gender was divided into two groups,
males and females. Age was divided into two groups,
Jpn J Compr Rehabil Sci Vol 8, 2017

24 Tokunaga M et al.: Increasing the prediction accuracy of multiple linear regression analysis for the prediction of FIM gain
69 years and younger, and 70 years and older, giventhat FIM improvement for patients aged 70 years andover generally decreases linearly with increasing age[14-16]. In terms of days from onset to admission, andcognitive FIM score at admission, there were twogroups for each of these factors, given that FIM
improvement differed if the number of days was 41
and below, or above 41, and if cognitive FIM score atadmission was 5–14 points or 15–35 points [17].
Given that the peak of motor FIM gain has been 25–30
points [14] and 31–36 points [17], motor FIM score atadmission was divided into two groups, 13–30 points
and 31–90 points, as in our previous report [11].
Median values were used for dividing mRS before
onset and number of hospital days (Table 1). mRS
before onset was divided into two groups, 0 points and
1–5 points, and number of hospital days was divided
into two groups, 99 days or less and 100 days or
longer. Thus, with these two groups (with the exceptionof three groups for stroke type), motor FIM gain wascompared. For a comparison between the two groups,
the Mann-Whitney U test was used. For a comparisonof the three groups, the Kruskal-Wallis test was
conducted, and if there was a statistical significance,
a multiple comparison was made using Scheffé’smethod.
Investigation 2: Multiple linear regression analysis
with motor FIM gain as the objective variable
As in our previous report [11], we conducted amultiple linear regression analysis with motor FIM
gain as the objective variable, and the five items ofage, mRS before onset, days from onset to admission,
motor FIM score at admission, and cognitive FIM
score at admission were the explanatory variables. At
this time, we created two prediction methods with thedivision of age into 69 years or younger and 70 years
or older. Similarly, for the other seven factors,
stratifications were made as in Investigation 1, withtwo prediction formulas created (three for stroke type).
Investigation 3: Stratification with the two factorsof motor FIM and cognitive FIM
In Investigation 2, the effect of the stratification ofeight factors was large for motor FIM score atadmission and cognitive FIM score at admission; thus,
we used these two factors for stratification. Specifically,
there were two groups for motor FIM score atadmission (13–30 points and 31–90 points), and twogroups for cognitive FIM score at admission (5–14
points and 15–35 points), for a total of four stratified
groups and four prediction formulas for the multiple
linear regression analysis.

Jpn J Compr Rehabil Sci Vol 8, 2017

Investigation 4: Motor FIM gain was predictedusing a multiple linear regression analysis bysubtracting motor FIM score at admission frommotor FIM score at discharge
In a report using a multiple linear regression analysis[1], given that a larger R2, coefficient of determination,
was gained with the prediction of motor FIM score atdischarge as opposed to the prediction of motor FIM
gain, we initially predicted motor FIM score atdischarge using multiple linear regression analysis.
Then, we subtracted motor FIM score at admission
from the predicted motor FIM score at discharge toobtain the predicted value for motor FIM gain.
We looked at the correlation between the predicted
and measured values for motor FIM gain obtainedin Investigations 2–4, and examined the difference
between the measured values and predicted values.
The correlation coefficient was obtained with thePearson correlation coefficient formula. The statisticalsoftware used was Statcel 4 [18] and Mulcel [19]. The
level of significance was set as < 5%.
Results
Statistical significance was observed with motor
FIM gain in the two stratified groups of age, mRS
before onset, days from onset to admission, motor
FIM score at admission, cognitive FIM score atadmission and number of hospital days (MannWhitney U test). Specifically, motor FIM gain waslarger in patients 69 years or younger, with mRS
before onset 0 points, less than 41 days from onset toadmission, motor FIM score at admission 30 points or
less, cognitive FIM score at admission 15 points or
higher, and number of hospital days 100 days or longer(Table 2). Statistical significance was also observed in
the three groups for stroke type (Kruskal-Wallis test),
and there was significantly higher motor FIM gain
with cerebral hemorrhage than with cerebral infarction(Scheffé’s method). In contrast, there was no clear
statistical significance with motor FIM gain according
to gender.
The partial regression coefficient for the multiple
linear regression analysis was a negative value for age,mRS before onset, days from onset to admission, andmotor FIM score at admission (the larger these values,
the smaller the motor FIM gain), and it was a positive
value for cognitive FIM score at admission (the higher
the cognitive FIM score at admission, the larger the
motor FIM gain) (Table 3).
The correlation coefficients for measured andpredicted values for motor FIM gain were, from largest
to smallest, 0.641 with the two-group stratificationaccording to motor FIM score at admission, 0.588
with the two-group stratification according to cognitive
FIM score at admission, 0.541 with the two-group
stratification according to age, 0.520 with the twogroup stratification according to mRS before onset,

Tokunaga M et al.: Increasing the prediction accuracy of multiple linear regression analysis for the prediction of FIM
gain
25

Table 2. Comparison of motor FIM gain according to stratified factors.
Stratification

Number of patients Motor FIM gain SignificanceAge

69 years and younger
70 years and older

1,133
1,409

21.8±15.5 (20)18.5±14.1 (16)p < 0.001

GenderMale
Female

1,492
1,050

20.0±14.6 (18)20.0±15.2 (18)0.68

Cerebral infarction
Cerebral hemorrhage
Subarachnoid hemorrhage

1,613
772
157

18.9±14.0 (17)*22.2±15.7 (20)*20.0±17.4 (16)p < 0.001

0
1‐5

1,795
747

20.7±15.0 (19)18.1±14.2 (16)p < 0.001

Days from onset to admission

41 days and below
Above 41 days

1,701
841

21.1±15.0 (19)17.7±14.3 (16)p < 0.001

Motor FIM score at admission

13‐30 points
31‐90 points

853
1,689

23.2±19.3 (20)18.3±11.7 (17)p < 0.01

Cognitive FIM score at admission

5‐14 points
15‐35 points

614
1,928

19.4±17.4 (16)20.1±13.9 (18)p < 0.01

99 days or less
100 days or longer

1,263
1,279

16.9±13.8 (14)22.9±15.2 (22)p < 0.001

Stroke typemRS before onset

Number of hospital days

Numerical value of motor FIM gain, Mean ± standard deviation (median);Significance, The Mann-Whitney U test was used for two-group comparisons, and the Kruskal-Wallis test wasused for the three-group comparison.
*, With a multiple comparison (Scheffé’s method), statistical significance was p < 0.01 between * and *.
0.518 with the two-group stratification according todays from onset to admission, 0.516 with the twogroup stratification according to number of hospitaldays, 0.512 with the three-group stratificationaccording to stroke type, 0.509 for the two-group
stratification according to gender, and 0.507 for a
prediction formula with no stratification (Table 3). The
average ‘residual,’ found by subtracting the predictedvalue from the measured value for motor FIM gain,
was 0, and the standard deviation (SD) of the residual
was in the range of 11.4 (motor FIM score at admission)
to 12.8 (gender).
In the four prediction formulas stratified by the two
factors of motor FIM score at admission and cognitive
FIM score at admission, the correlation coefficient
between the measured values and predicted values formotor FIM gain was 0.653, the highest correlation
coefficient (Table 4). The residual (average ± SD) was
0±11.2; therefore, the SD of the residual was thelowest. In terms of the partial regression coefficient for
motor FIM score at admission, for the 13–30 point
group it was a positive value (the higher the motor
FIM score at admission, the larger the motor FIMgain), and for the 31–90 point group it was a negativevalue (the higher the motor FIM score at admission,
the smaller the motor FIM gain). With regard to thesefour prediction formulas, if the measured value formotor FIM gain is on the horizontal axis, and thepredicted value is on the vertical axis (Fig. 3c, d), thevalues are slightly closer to the linear curve in which
the measured and predicted values for motor FIM gain
meet, compared to the non-stratified single prediction
formulas (Fig. 3a, b). However, it was difficult to makepredictions for patients with motor FIM score atadmission of 13–30 points (Fig. 3c) and for patientswith measured motor FIM gain values of both
extremes, close to 0 or to 60 points.
In multiple linear regression analysis to predict
motor FIM score at discharge, R2 was 0.693. If thepredicted value for motor FIM gain is obtained using
multiple linear regression analysis by subtracting
motor FIM score at admission from the predictedvalue for motor FIM score at discharge, the correlation
coefficient for the measured value and predicted valueof motor FIM gain becomes 0.507, the same value as
directly obtained with multiple linear regression
analysis.
Discussion
The correlation coefficient for the measured value
and predicted value of motor FIM gain was 0.507 witha single prediction formula; however, it was 0.641 if
two prediction formulas were created with motor FIM
score at admission. This is the same result as in ourprevious study [11]. In the present study, weinvestigated the significance of stratifying the sevenfactors of age, gender, stroke type, mRS before onset,
Jpn J Compr Rehabil Sci Vol 8, 2017

26 Tokunaga M et al.: Increasing the prediction accuracy of multiple linear regression analysis for the prediction of FIM gain
Table 3. Multiple linear regression analyses to predict motor FIM gain.
Motor FIM score atadmission
Stratification

No.

Age

GenderStroke type

13–30
points31–90
points69 years
andyounger

70 years
and
older

Male

Female

2,542

853

1,689

1,133

1,409

1,492

1,050

1,613

772

157

−0.212−1.528−0.45−1.493−0.142−1.233−0.130−1.123−0.300−1.553−0.178−1.486−0.265−1.517−0.179−1.404−0.229−1.682−0.160−0.170−0.139−0.211−0.092−0.084−0.189−0.146−0.128−0.130−0.171−0.166−0.4100.674−0.561−0.505−0.326−0.410−0.408−0.400−0.425−0.439

0.539

0.653

0.194

0.479

0.575

0.506

0.578

0.551

0.583

0.461

Constants
p ValueR2

47.64743.57259.89947.19551.89346.21850.22243.711
p < 0.001 p < 0.001 p < 0.001 p < 0.001 p < 0.001 p < 0.001 p < 0.001 p < 0.001
0.257
0.314
0.510
0.366
0.203
0.259
0.260
0.248

49.950p < 0.001
0.273

50.736p < 0.001
0.227

Correlation
Residual

0.507
0±12.8

Number of patients
Explanatory variables
AgemRS before onset
Days from onset toadmission
Motor FIM scoreat admission
Cognitive FIM scoreat admission

0.641
0±11.4mRS before onset
Stratification

Number of patients
Explanatory variables
AgemRS before onset
Days from onset toadmission
Motor FIM scoreat admission
Cognitive FIM scoreat admission
Constants
p Value
R2
Correlation
Residual

0.541
0±12.5

Cerebral
Cerebral Subarachnoid
infarction hemorrhage hemorrhage

0.509
0±12.8Days from onset toadmission

0.512
0±12.7

Cognitive FIM score atadmission

Number of hospital days

0

1‐5

41days andbelow

Above 41days

5‐14 points

15‐35 points

99 days orless

100 days or
longer

1,795

747

1,701

841

614

1,928

1,263

1,279

−0.188/−0.293−1.236−0.218−1.554−0.204−1.514−0.404−1.417−0.184−1.398−0.141−1.213−0.272−1.771−0.137−0.149−0.121−0.061−0.155−0.131−0.14−0.147−0.441−0.304−0.454−0.318−0.151−0.461−0.397−0.381

0.509

0.580

0.521

0.536

2.111

0.299

0.489

0.578

48.115p < 0.001
0.301

48.524p < 0.001
0.172

50.218p < 0.001
0.303

38.753p < 0.001
0.164

40.068p < 0.001
0.247

54.608p < 0.001
0.395

42.355p < 0.001
0.241

50.960p < 0.001
0.230

0.520
0±12.70.518
0±12.70.588
0±12.00.516
0±12.7

Numerical value of explanatory variables, Partial regression coefficient; Numerical value of the residual, Mean ± standard deviation;
R2, Coefficient of determination;
Correlation, Correlation between measured value and predicted value of motor FIM gain (Pearson correlation coefficient);
Residual, Residual error obtained by subtracting the predicted motor FIM gain from the measured motor FIM gain;
The results of no stratification and stratification of motor FIM score at admission were the same as in Ref. 11.
/, If mRS prior to onset was 0, mRS prior to onset was not added to the explanatory variables.

days from onset to admission, cognitive FIM score atadmission, and number of hospital days. The resulting
correlation coefficients were, in order of size, 0.588
for cognitive FIM score at admission, 0.541 for age,
0.520 for mRS before onset, 0.518 for days fromonset to admission, 0.516 for number of hospital days,
and 0.512 for stroke type. The correlation coefficientfor gender did not exceed 0.509, and given that there
was no significance in terms of gender even in the
Jpn J Compr Rehabil Sci Vol 8, 2017

comparison of motor FIM gain in Investigation 1, we
believe that there is no need to create separate
prediction formulas for males and females.
Of the five explanatory variables for the multiple
linear regression analysis, for the two stratified
prediction formulas for age, mRS before onset, days
from onset to admission and cognitive FIM score atadmission, the partial regression coefficient values forthese factors differed. However, the mathematical sign

Tokunaga M et al.: Increasing the prediction accuracy of multiple linear regression analysis for the prediction of FIM
gain
27

Table 4. Four prediction formulas stratified by motor FIM and cognitive FIM.
Motor FIM score at admission
Cognitive FIM score at admission

13–30 points

31–90 points

5–14 points

15–35 points

5–14 points

15–35 points

Number of patients

482

371

132

1,557

Explanatory variables
AgemRS before onset
Days from onset to admission
Motor FIM score at admission
Cognitive FIM score at admission−0.429−1.186−0.1670.781
1.613−0.499−1.740−0.2520.367
0.709−0.276−2.502−0.063−0.4780.655−0.132−1.134−0.094−0.566
0.203

Constants
p ValueR2

30.370p < 0.001
0.314

52.657p < 0.001
0.262

59.714p < 0.001
0.333

59.311
p < 0.001
0.529

Correlation
Residual

0.653
0±11.2For example, for a patient with motor FIM score at admission of 13 points, and cognitive FIM score atadmission of 5 points, the leftmost formula is used to calculate the predicted value of motor FIM gain.

Figure 3. Relationship between measured value and predicted value of motor FIM gain.
Horizontal axis, Measured value of motor FIM gain; Vertical axis, Predicted value ofmotor FIM gain; Dot, each patient.
Stratified formula, Four prediction formulas stratified by the two factors of motor FIM
score at admission and cognitive FIM score at admission.
was the same. For example, the partial regression
coefficient for cognitive FIM score at admission was
2.111 in the 5–14 point group, and 0.299 in the 15–35
point group. In contrast, the sign for the partial

regression coefficient for motor FIM score at admission
was positive in the 13–30 point group, and negative inthe 31–90 point group, which was accurately reflectedin the mountain-shaped relationship (Fig. 1) between
Jpn J Compr Rehabil Sci Vol 8, 2017

28 Tokunaga M et al.: Increasing the prediction accuracy of multiple linear regression analysis for the prediction of FIM gain
motor FIM score at admission and motor FIM gain.
Thus, we believe that stratification is necessary in the
explanatory variables, which has no linear relationshipwith the objective variable.
Meyer et al. [1], who reviewed the multiple linear
regression analysis, stated that “Interestingly,
admission FIM scores were always positively
correlated with higher discharge FIM scores; however,results for FIM gain were mixed. Admission FIM
score was negatively correlated with FIM gain in sixof the nine models.” This can be explained as follows:
Since the number of patients with motor FIM score atadmission of 31–90 points (negative partial regression
coefficient for motor FIM score at admission) was
higher than the number of patients with 13–30 points(positive partial regression coefficient), if only one
prediction formula is created, the partial regression
coefficient for motor FIM score at admission becomesa negative value.
Given that there was great significance for stratifying
motor FIM score at admission and cognitive FIM
score at admission, four prediction formulas werecreated with the combination of these two factors. The
correlation coefficient became the highest (0.653) andthe SD of the residual became the lowest. For thatreason, in the multiple linear regression analysis
conducted in this study, it was considered that thehighest prediction accuracy for motor FIM gain wasachieved with this method of ‘stratifying motor FIM
and cognitive FIM, and creating four prediction
formulas.’ Creating one prediction formula appropriate
for all stroke patients is difficult. Therefore, multiple
formulas should be created to match the patient’s
motor FIM and cognitive FIM.
Furthermore, we considered eight-group stratification
by adding age, which was the third necessity ofstratification, to motor FIM score at admission andcognitive FIM score at admission. However, given that
there were 132 patients in the group with motor FIM
score at admission of 31–90 points and with cognitive
FIM score at admission of 5–14 points (Table 4), wewere not able to further stratify the group using age.
This is because in multiple linear regression analysis, atleast 75 patients are required in each group if fiveexplanatory variables are used, since it is necessary tohave the following: number of explanatory variables ×
15 or more patients [20].
Given that the coefficient of determination or R2(average 0.65) used for predicting FIM score atdischarge was larger than the R2 (average 0.22) used
for predicting FIM gain, in the multiple linear
regression analysis [1], motor FIM score at discharge
was predicted first. Then, by subtracting motor FIM
score at admission from this, we tried to obtain apredicted value for motor FIM gain. However, theprediction accuracy for motor FIM gain was the sameas directly obtained with multiple linear regression
analysis. Given that it is “FIM score at discharge=FIM
Jpn J Compr Rehabil Sci Vol 8, 2017score at admission+FIM gain,” there was a correlation
between FIM score at admission and FIM score atdischarge. For that reason, R2 for predicting FIM scoreat discharge was larger. So, R2 for FIM score atdischarge (the same as R2 for FIM gain) cannot be said
to be a high value.
This study had the following limitations. First, it
was difficult to predict the functional outcome forpatients with motor FIM score at admission of 13–30
points. While the average residual was 0 points,
SD was 11.2–12.8. In particular, making accurate
predictions for patients with motor FIM gain of 0
points (with no improvement whatsoever) and forpatients with motor FIM gain of close to 60 points wasdifficult. Second, there was the question of whether
five explanatory variables were sufficient or not.
However, even in Meyer et al.’s review [1], the numberof significant explanatory variables included in theprediction formulas was 4.1 on average. Third, there is
the possibility of other factors that should be stratified
besides the factors considered in our study. Fourth,
validity was not examined. It would have been
desirable to divide the target patients into two groups,
a creation group and a verification group, and to
examine the intrinsic validity of the prediction
formulas using the verification group’s data, or to
examine the external validity using data from otherfacilities [21]. However, even though our target
patients numbered 2,542, when motor FIM score andcognitive FIM score at admission were stratified into
four groups, the smallest group became only 132
patients. If the target patients had been divided intocreation and verification groups, it would have been
about 66 patients. At this level, Investigation 3 could
not have been conducted. Fifth, if there were further
stratifications, the number of patients per group would
decrease, and analysis would become cumbersome.
Lastly, there is the possibility that the results willchange depending on how detailed the stratification isand how many groups there are.
As a method for increasing the prediction accuracyof multiple linear regression analysis, in addition to
the stratification investigated in our study, there is also
the option of having an appropriate number ofappropriate explanatory variables. In Meyer et al.’sreview of 27 reports on multiple linear regression
analysis of acute stroke patients [1], 126 factors wereused as explanatory variables, and of those, there were
63 factors with significance. The Japanese Guidelines
for the Management of Stroke 2015 [22] states thatprediction accuracy will not necessarily increase bysimply increasing the number of explanatory variables
used for prediction. However, there are some reports
stating that prediction accuracy can be increased
with the addition of explanatory variables such ascomorbidities [23], Stroke Impairment Assessment
Set [3, 24], and the median FIM gain according to FIM
score at admission [11].

Tokunaga M et al.: Increasing the prediction accuracy of multiple linear regression analysis for the prediction of FIM
gain
29

An issue for future research is accurate prediction
for patients with FIM gain of 0 points and for patientswith extremely high FIM gain.
Acknowledgements
This study used the Japan Rehabilitation Databasemanaged by the Japan Association of Rehabilitation
Database, to whom we extend our gratitude. The
contents and conclusions of this study are not theopinions of the Association, but rather the viewpointsof the authors.

References

1. Meyer MJ, Pereira S, McClure A, Teasell R, Thind A,
Koval J, et al. A systematic review of studies reporting
multivariable models to predict functional outcomes after
post-stroke inpatient rehabilitation. Disabil Rehabil 2015;
37: 1316-23.
2. Inouye M. Predicting models of outcome stratified byage after first stroke rehabilitation in Japan. Am J Phys
Med Rehabil 2001; 80: 586-91.
3. Sonoda S, Saitoh E, Domen K, Chino N. Prognosticationof stroke patients using SIAS and FIM. In: Chino N,
Melvin JL, editor. Functional Evaluation of Stroke
Patients. Tokyo: Springer-Verlag; 1995. p.103-14.
4. Hirano Y, Okura Y, Takeuchi M. The influence of ADL
severity at admission on ADL at discharge in convalescentstroke rehabilitation. Tohoku Rigaku-ryoho Kagaku 2011;
23: 32-7. Japanese.
5. Tokunaga M, Taniguchi M, Nakakado K, Mihono T,
Okido A, Ushijima T, et al. Assessment of the effects offactors in stroke rehabilitation using eight multiple
regression analyses. Jpn J Compr Rehabil Sci 2015; 6:
78-85.
6. Wang H, Camicia M, Terdiman J, Hung YY, Sandel ME.
Time to inpatient rehabilitation hospital admission andfunctional outcomes of stroke patients. PM & R 2011; Apr3(4): 296-304.
7. Tokunaga M, Kondo K. Training time and FIM gain inpatients with stroke in Kaifukuki rehabilitation hospitals.
Sogo Rehabil 2014; 42: 245-52. Japanese.
8. Imada Y, Tokunaga M, Fukunaga K, Sannomiya K,
Inoue R, Hamasaki H, et al. Relationship between
cognitive FIM score and motor FIM gain in patients withstroke in a Kaifukuki rehabilitation ward. Jpn J Compr
Rehabil Sci 2014; 5: 12-8.
9. Tokunaga M, Ikeda Y, Inoue M, Kodama J, Sakamoto
M, Nagatomo M, et al. Multiple regression analysis
stratified by age and FIM at admission. J Clin Rehabil
2015; 24: 828-34. Japanese.
10. Sonoda S, Nagai S, Saitoh E. A viewpoint and problem

of the convalescent rehabilitation wards. Jpn J Rehabil
Med 2005; 42: 614-7. Japanese.
11. Tokunaga M, Mori Y, Ogata Y, Tanaka Y, Uchino K,
Maeda Y, et al. Predicting FIM gain in stroke patients byadding median FIM gain stratified by FIM score at
hospital admission to the explanatory variables in multiple
regression analysis. Jpn J Compr Rehabil Sci 2016;
7: 13-8.
12. Kondo K. Secondary analysis of the rehabilitation
patient database. Jpn J Rehabil Med 2012; 49: 142-8.
Japanese.
13. Japan Association of Rehabilitation Database. Japanese.
Available from: http://square.umin.ac.jp/JARD/
14. Tokunaga M, Nakanishi R, Eguchi G, Kihara K, Tokisato
K, Katsura K, et al. The influence of age on corrected
motor FIM effectiveness. Jpn J Compr Rehabil Sci 2014;
5: 56-60.
15. Tokunaga M, Watanabe S, Nakanishi R, Yamanaga H,
Yonemitsu H, Mita S, et al. The influence of stroke type,gender, and age on FIM improvement. Jpn J Compr
Rehabil Sci 2014; 5: 136-40.
16. Tokunaga M, Honda S, Uyama E, Mita S, Terasaki T,
Hirata Y, et al. Difference of FIM improvement between
subtypes of brain infarction. J Clin Rehabil 2014; 23:1236-40. Japanese.
17. Tokunaga M, Sannomiya K, Nakashima Y, Nojiri S,
Tokisato K, Katsura K, et al. Formula for predicting FIM
gain and discharge FIM: methods using median values of
FIM gain stratified by admission FIM, age, cognitive
function, and transfer interval. Jpn J Compr Rehabil Sci
2015; 6: 6-13.
18. Yanai H. 4 Steps Excel Statistics. Saitama: OMS
Publishing Inc; 2005. p. 1-309. Japanese.
19. Yanai H. Practical Multivariate Analysis Greate Utilities
on Excel. Saitama: OMS Publishing Inc; 2005. p. 1-174.
Japanese.
20. Shintani A. Medical Statistics. Tokyo: Igaku-shoin;2015. p.1-167. Japanese.
21. Tokunaga M, Sannomiya K, Nakanishi R, Yonemitsu H.
The external validity of multiple regression analyses
predicting discharge FIM score in patients with stroke
hospitalized in Kaifukuki rehabilitation wards. Jpn J
Compr Rehabil Sci 2015; 6: 14-20.
22. The Japan Stroke Society. Japanese Guidelines for theManagement of Stroke 2015. Tokyo: Kyowa Kikaku;
2015. p.1-337. Japanese.
23. Liu M, Domen K, Chino N. Comorbidity measures forstroke outcome research. Arch Phys Med Rehabil 1997;
78: 166-72.
24. Tsuji T, Liu M, Sonoda S, Domen K, Chino N. The
stroke impairment assessment set. Arch Phys Med Rehabil
2000; 81: 863-8.

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