The lengths of spinal curvature stretch due to the angles of sitting on saddle chair to alleviate back pain: A statistical analysis

Abstract

BACKGROUND:

By dividing the burden of one’s weight between the shins and the buttocks in the sitting position on an office or saddle chair, a person can avoid back pain. In this 21^st century, sitting on a chair for long hours in workplace on office chair is unavoidable necessity and hence, millions in different countries undergo a risk for backpain. Is there a right sitting position?

OBJECTIVE:

The aim of this article is to find out how much a correlation exists between the angle of sitting and the length of spinal curvature which is the source of backpain. An experiment can be designed and carried out to measure various angles in sitting and the changing length of the person’s spinal cord curvature.

METHOD:

The usual statistical methodology requires a pair of values namely x and y to quantify the correlation. The data on sitting angles and the length of spinal curvature do not have such pairing, and hence, the traditional approach to find the correlation between the sitting angle and length of spinal curvature is not applicable. Yet, an approach is necessary. This article constructs an innovative statistical approach to fulfil this need.

RESULTS:

Our approach yields a correlation of 0.998 for sitting on office chair and an increased correlation of 0.999 on saddle chair, according to the Truszczyńska-Baszaka et al.’s data.

CONCLUSIONS:

An adjustment is made in various angles of sitting on office chair to transform the comfortable sitting on a saddle chair. In consequence, the proportional effect on the spinal curvature is estimable with the data and it is phenomenal (that is significantly more than one). No wonder people prefer saddle chair over office chair when it comes to avoid back pain and this article proves the convenience statistically.

Keywords

Bivariate distribution linear model conditional mean and variance Kronecker vector

1 Introduction

The traditional statistical method to find the correlation between two random variables x and y requires them to be in pairs. This requirement is not met at times in physical therapy studies as the one involved in this article. A case in point is that the data on sitting angles and the length of spinal curvature, which do not have such pairing. Hence, the traditional approach to find the correlation between the sitting angle and length of spinal curvature is not applicable. Yet, an approach is necessary. This article constructs an innovative statistical approach to fulfil this need.

A research question exists in health research activities. That is how to find correlation when the data do not have a pair of matching measurements. This research article constructs an innovative methodology to quantify the correlation in such a situation. Using the derived expressions in this article, we obtain an estimate of the correlation between the angles of sitting and lengths of spinal curvature using data from Truszczyńska-Baszaka et al. [4]. We noticed that the correlation is negative, meaning that the length of the spinal curvatures is inversely related to the sitting angle.

Physical therapists often recommend that back pain is avoidable by proper sitting angles (see Wang [5] for details). The back pain is not caused only by the spinal deformities (kyphosis, lordosis, scoliosis) but also by the congenital disorders, disc hernia etc. To understand more about the prevalence of the back pain, Truszczyńska-Baszaka et al. [4] recently measured and evaluated changes in ten spinal curvatures due to changing seven angles of sitting positions on office versus saddle chair. They reported that back pain had been significantly reduced by sitting on saddle chair. In their report, the seven angles of sitting are inputs and ten lengths of spinal curvature are outcome variables in the physiologic system. One wonders what the physiologic system’s correlation between the input and output variables might be. The usual formula to find correlation is not applicable for a lack of match between the input and output variables. Yet, an approach is necessary. This article constructs a statistical approach to fulfil this need. Using the derived expressions in our construct, we obtain an estimate of the system’s correlation between the angles of sitting and lengths of spinal curvature. The data in Truszczyńska-Baszaka et al. [4] are utilized to illustrate our new methodology. Our finding shows that the physiologic system’s correlation is negative, meaning that the length of the spinal curvatures is inversely related to the sitting angle. The use of saddle chair eases the back pain by significantly altering several spinal curvatures, as demonstrated by this article.

The spine, or backbone as it is popularly recognized, is made up of small bones (vertebrae) stacked along with discs one on top of another. Spine is a vital organ for a happy and healthy life. A healthy spine has gentle curves to it. The curves help the spine to manage stress optimally from the body movements and natural gravity. When abnormalities occur in the spine, its curvature is misaligned, and it causes back pain. There are three types of spine disorders, and it includes Lordosis (referring swayback), Kyphosis (referring abnormally rounded upper back with more than 50 degrees of curvature), Scoliosis (referring leaning with S- or C-shape).

Sitting on office chair for a long hour, unfortunately, causes back pain. Needless to point out that numerous full-time employees in offices or other places undergo such back pain. Alternatively, sitting on saddle chair with angular changes offers spinal mobility. The angular changes in the spine (because of the flexibility in forward incline, recline, and upright) perhaps make a healthy reduction in back pain. To check out this line of thinking, Truszczyńska-Baszaka et al. [4] recently performed a physiotherapy experiment and collected pertinent data as redisplayed in Table 1 by involving 60 randomly selected healthy able-bodied students (23 men and 37 women). They considered seven input variables Alpha, (angle between the greatest depth of lumbar lordosis and the base of the sacrum), Beta (angle between the largest value of thoracic kyphosis and the thoracolumbar transition), Gamma (angle between a section and the greatest value of thoracic kyphosis), KKP (180–(BETA + GAMMA)), KLL (180–(ALPHA + BETA)), KNT (Deviation (leaning forward/backward from the vertical line in the frontal plane) and KPT (deviation from a vertical line in the sagittal plane) measured in degrees. The ten output variables are DKP (length on the top of thoracic kyphosis), GKP (depth from the top of thoracic kyphosis to the thoracolumbar transition), GLL (parameter from the greatest depth of lumbar lordosis to the thoracolumbar transition), KLB (inclination of the line joining the inferior angles of the shoulder blades relative to the horizontal line), KNM (inclination of the line connecting the posterior superior iliac spines to the horizontal line), KSM (inclination of the line connecting the posterior superior iliac spines relative to the sagittal plane), UB (deviation from the horizontal line of the line joining the inferior angles of the shoulder blades in the sagittal plane), UK (place of the maximum deviation of the spinous process on the line), UL (deviation from the horizontal line of the line joining the inferior angles of the shoulder blades) measured in milli meter (mm) and OL (sections between consecutive spinous processes approximated with straight lines) measured in percent (%).

Table 1
Angle (x) and length (y) of curvature in sitting on office (x_o, y_o and saddle chair (x_s, y_s)

Input variable x _o x _s Output variable y _o y _s

Alpha deg 25.70 18.8 DKP mm 171.80 167.9

Beta deg 3.10 3.4 GKP mm 0.80 –1.3

Gamma deg 26.40 33.3 GLL mm –0.20 1.5

KKP deg 152.90 147.3 KLB mm 1.60 0.9

KLL deg 197.20 188.6 KNM mm 0.00 0.2

KNT deg –0.50 –0.5 KSM mm 4.00 3.3

KPT deg –4.80 –12.3 OL % 1.00 1.2

UB mm 3.00 3.3

UK mm 6.00 –0.2

UL mm 0.30 0.9

Average 57.14 54.08 Average 18.83 17.77

Variance 6799.91 6402.63 Variance 2892.76 2784.61

Correlation _office 0.998 Correlation _saddle 0.999

Input variable	x _o	x _s	Output variable	y _o	y _s
Alpha deg	25.70	18.8	DKP mm	171.80	167.9
Beta deg	3.10	3.4	GKP mm	0.80	–1.3
Gamma deg	26.40	33.3	GLL mm	–0.20	1.5
KKP deg	152.90	147.3	KLB mm	1.60	0.9
KLL deg	197.20	188.6	KNM mm	0.00	0.2
KNT deg	–0.50	–0.5	KSM mm	4.00	3.3
KPT deg	–4.80	–12.3	OL %	1.00	1.2
			UB mm	3.00	3.3
			UK mm	6.00	–0.2
			UL mm	0.30	0.9
Average	57.14	54.08	Average	18.83	17.77
Variance	6799.91	6402.63	Variance	2892.76	2784.61
Correlation _office	0.998		Correlation _saddle	0.999

The article is organized as follows. In Section 2, the main methodology to analyze the collected data to capture the non-observed correlation between the input and output variables of the physiology system. In Section 3, an illustration and an interpretation are made. A few comments and conclusions are made in Section 4 in the end.

2 New methodology based on multivariate linear model

To construct an appropriate methodology to quantify the correlation in data with nonmatching variables, we consider the following symbols and notations. The input and out variables are indicated by x and y respectively. Notice that there might be two groups of variables like in the example of sitting on office or saddle chairs with back pain. Each (input as well as output) variable is linked between the two groups of the system as in the physiologic example.

Let x_o and x_s denote respectively their measured angle sitting in the office chair or saddle chair. We assume that there exists an intrinsic relation x_s = θx_o (see Fig. 1), where both components in the vector $\underline{x} = (\begin{matrix} x_{o} \\ x_{s} \end{matrix})$ are Gaussian distributed (see the theoretical versus observed quantile-quantile plot of sitting angle in office chair, x_o in Fig. 2 and sitting angle in saddle chair, x_s in Fig. 3) random variables (rv).

Fig. 1

The intrinsic relation x_s = θx_o.

Fig. 2

Sitting angle in office chair x_o.

Fig. 3

Sitting angle in saddle chair x_s.

In other words, $\underline{x} \sim N_{2} ({\underline{m}}_{x} μ_{x}, {{\underline{m}}_{x} \otimes {\underline{m}}_{x}} σ_{x}^{2})$ , where $(\begin{matrix} a \\ b \end{matrix}) \otimes (\begin{matrix} c & d \end{matrix}) = (\begin{matrix} ac & ad \\ bc & bd \end{matrix})$ is called Kronecker matrix product (see Tracy and Singh, [3] for details), ${\underline{m}}_{x} = [\begin{matrix} 1 \\ 1 + θ \end{matrix}]$ , μ_x = E (x) is the expected value and $σ_{x}^{2} = Var (x)$ . Similarly, we assume for the output variable vector $\underline{y} = (\begin{matrix} y_{o} \\ y_{s} \end{matrix})$ with an intrinsic relation is y_s = ϕy_o. We assume that $\underline{y}$ follows a bivariate Gaussian frequency pattern (see quantile-quantile plots in Figs. 4 and 5).

Fig. 4

Curvature for office chair y_o.

Fig. 5

Curvature for saddle chair y_s.

In other words, $\underline{y} \sim N_{2} ({\underline{m}}_{y} μ_{y}, {{\underline{m}}_{y} \otimes {\underline{m}}_{y}} σ_{y}^{2}),$ where μ_y = E (y) is the expected value and $σ_{y}^{2} = Var (y)$ is the variance. The system’s inputs and outputs are not necessarily independent and let their dependency is indicated by ρ. To capture an expression to estimate ρ, there is no appropriate methodology in the literature. Hence, this research article aims at devising an approach and derive appropriate expressions.

For this purpose, we stack up the input and output vectors as a vector $\underline{z} = (\begin{matrix} \underline{x} \\ \underline{y} \end{matrix})$ and realize that it follows a multivariate Gaussian distribution

$\begin{matrix} N_{4} ((\begin{matrix} {\underline{m}}_{x} μ_{x} \\ {\underline{m}}_{y} μ_{y} \end{matrix}), \\ (\begin{matrix} {{\underline{m}}_{x} \otimes {\underline{m}}_{x}} σ_{x}^{2} & {{\underline{m}}_{x} \otimes {\underline{m}}_{y}} ρ σ_{x} σ_{y} \\ {{\underline{m}}_{x} \otimes {\underline{m}}_{y}} ρ σ_{x} σ_{y} & {{\underline{m}}_{y} \otimes {\underline{m}}_{y}} σ_{y}^{2} \end{matrix})) \end{matrix}$ (1)

Note that when the correlation, ρ is zero, the input and output variables become independent with no link, because the zero covariance characterizes the independence only among the multivariate Gaussian rv not necessarily for any non-multivariate populations (see Srivastava, [2] for details). Equivalently mentioning, for there to be dependency between the input and output vectors, the covariance matrix in (1) must be non-singular and it means that the determinant of the covariance matrix in (1) must be non-zero. From the non-zero determinant, we obtain an expression for the system’s correlation $ρ_{x, y} = \sqrt{\frac{| {{\underline{m}}_{x} \otimes {\underline{m}}_{x}} {{\underline{m}}_{y} \otimes {\underline{m}}_{y}} |}{| {[{{\underline{m}}_{x} \otimes {\underline{m}}_{y}} {{\underline{m}}_{y} \otimes {\underline{m}}_{x}}]}^{- 1} |}}$ (2)

where the symbol $| \underline{B} |$ is the determinant of the matrix $\underline{B}$ . The maximum likelihood estimate (MLE) of the dependency measure in (2) between the system’s input $(\underline{x})$ and output $(\underline{y})$ variables is obtained using the MLEs ${\hat{θ}}_{mle} = \frac{{\bar{x}}_{o}}{{\bar{x}}_{s}} - 1$ and ${\hat{ϕ}}_{mle} = \frac{{\bar{y}}_{o}}{{\bar{y}}_{s}} - 1$ (see the estimating space in Fig. 6).

Fig. 6

Estimating space.

It is so because of the invariance property of the MLE (that is, the MLE of a function of the parameters is the function of the MLE of the parameters). The MLEs are ${\hat{\underline{m}}}_{x, mle} = (\begin{matrix} 1 \\ {\hat{θ}}_{mle} \end{matrix})$ and ${\hat{\underline{m}}}_{y, mle} = (\begin{matrix} 1 \\ {\hat{ϕ}}_{mle} \end{matrix})$ . Within the input vector in the first layer, the sample means ${\bar{x}}_{o}$ and ${\bar{x}}_{s}$ are not independent. Likewise, the sample means ${\bar{y}}_{o}$ and ${\bar{y}}_{s}$ are not independent within the output vector in its first layer.

Consequently, the MLE ${\hat{θ}}_{mle}$ is convergingly unbiased, since $E ({\hat{θ}}_{mle}) = \frac{E ({\bar{x}}_{o})}{E ({\bar{x}}_{s})} - \frac{cov ({\bar{x}}_{o}, {\bar{x}}_{s})}{{E ({\bar{x}}_{s})}^{2}} - 1$ , (see Blumenfeld, [1] for details), where the Bias_θ = - θ (1 + [1 + θ] ^-1) →0, when the angle’s shift between the office and saddle chair is nullified (that is, θ → 0).

Likewise, we could obtain an expression for the variance and it is $\begin{matrix} Var ({\hat{θ}}_{mle}) = Var (\frac{{\bar{x}}_{o}}{{\bar{x}}_{s}} - 1) \\ = {(\frac{E ({\bar{x}}_{o})}{E ({\bar{x}}_{s})})}^{2} (| \begin{matrix} \frac{Var ({\bar{x}}_{o})}{{(E ({\bar{x}}_{o}))}^{2}} + \frac{Var ({\bar{x}}_{s})}{{(E ({\bar{x}}_{s}))}^{2}} \\ - \frac{2 cov ({\bar{x}}_{o}, {\bar{x}}_{s})}{E ({\bar{x}}_{o}) E ({\bar{x}}_{s})} \end{matrix} |) \\ = \frac{2 θ σ_{o, x}^{2}}{{(1 + θ)}^{2} μ_{o, x}^{2}} \end{matrix}$

(see Blumenfeld, 2010 for details). The mean square (MS) of the MLE ${\hat{θ}}_{mle}$ is $Var ({\hat{θ}}_{mle}) + {Bias ({\hat{θ}}_{mle})}^{2} \to 0$ , when θ → 0 implying that the estimate is MS efficient. Proceeding likewise, we note that the MS of the MLE ${\hat{ϕ}}_{mle}$ is $\frac{2 ϕ σ_{o, y}^{2}}{{(1 + ϕ)}^{2} μ_{o, y}^{2}} + {(ϕ (1 + {[1 + ϕ]}^{- 1}))}^{2} \to 0$ , when the shift in the length of the spinal curvature between the office and saddle chair is nullified (that is, ϕ → 0) implying that this estimate is also MS efficient. An asymptotic confidence interval for each parameter is therefore possible. That is, $θ \in {\hat{θ}}_{mle} \pm z_{α / 2} \sqrt{\begin{matrix} {\frac{2 {\hat{θ}}_{mle} s_{o, x}^{2}}{{(1 + {\hat{θ}}_{mle})}^{2} {\bar{x}}_{o, x}^{2}} \\ + {[{\hat{θ}}_{mle} (1 + {[1 + {\hat{θ}}_{mle}]}^{- 1})]}^{2}} \end{matrix}}$ and

$θ \in {\hat{ϕ}}_{mle} \pm z_{α / 2} \sqrt{\begin{matrix} {\frac{2 {\hat{ϕ}}_{mle} s_{o, y}^{2}}{{(1 + {\hat{ϕ}}_{mle})}^{2} {\bar{x}}_{o, y}^{2}} \\ + {[{\hat{ϕ}}_{mle} (1 + {[1 + {\hat{ϕ}}_{mle}]}^{- 1})]}^{2}} \end{matrix}}$

3 Illustration of spinal curvature versus sitting angles

In this section, we illustrate the data in Table 1. Note that ${\hat{θ}}_{mle} = \frac{{\bar{x}}_{o}}{{\bar{x}}_{s}} - 1 = 0.056$ , ${\hat{ϕ}}_{mle} = \frac{{\bar{y}}_{o}}{{\bar{y}}_{s}} - 1 = 0.059$ , $\begin{matrix} {{\hat{\underline{m}}}_{x} \otimes {\hat{\underline{m}}}_{x}} \\ = (\begin{matrix} 1 & (1 + {\hat{θ}}_{mle}) \\ (1 + {\hat{θ}}_{mle}) & {(1 + {\hat{θ}}_{mle})}^{2} \end{matrix}) \\ = (\begin{matrix} 1 & 1.056 \\ 1.056 & 1.115 \end{matrix}), \end{matrix}$ $\begin{matrix} {{\hat{\underline{m}}}_{y} \otimes {\hat{\underline{m}}}_{y}} \\ = (\begin{matrix} 1 & (1 + {\hat{ϕ}}_{mle}) \\ (1 + {\hat{ϕ}}_{mle}) & {(1 + {\hat{ϕ}}_{mle})}^{2} \end{matrix}) \\ = (\begin{matrix} 1 & 1.059 \\ 1.059 & 1.121 \end{matrix}), \end{matrix}$ $\begin{matrix} {{\hat{\underline{m}}}_{x} \otimes {\hat{\underline{m}}}_{x}} {{\hat{\underline{m}}}_{y} \otimes {\hat{\underline{m}}}_{y}} = \\ (\begin{matrix} 1 + (1 + {\hat{θ}}_{mle}) (1 + {\hat{ϕ}}_{mle}) & (1 + {\hat{ϕ}}_{mle}) {1 + \\ (1 + {\hat{θ}}_{mle}) (1 + {\hat{ϕ}}_{mle})} \\ (1 + {\hat{θ}}_{mle}) {1 + & (1 + {\hat{θ}}_{mle}) (1 + {\hat{ϕ}}_{mle}) {1 \\ (1 + {\hat{θ}}_{mle}) (1 + {\hat{ϕ}}_{mle})} & + (1 + {\hat{θ}}_{mle}) (1 + {\hat{ϕ}}_{mle})} \end{matrix}) \\ = (\begin{matrix} 2.119 & 1.123 \\ 2.239 & 2.372 \end{matrix}), \end{matrix}$ and $\begin{matrix} {{\hat{\underline{m}}}_{y} \otimes {\hat{\underline{m}}}_{x}} {{\hat{\underline{m}}}_{x} \otimes {\hat{\underline{m}}}_{y}} \\ = (\begin{matrix} 1 + {(1 + {\hat{ϕ}}_{mle})}^{2} & (1 + {\hat{θ}}_{mle}) {1 + \\ {(1 + {\hat{ϕ}}_{mle})}^{2}} \\ (1 + {\hat{θ}}_{mle}) {1 + & {(1 + {\hat{θ}}_{mle})}^{2} {1 + \\ {(1 + {\hat{ϕ}}_{mle})}^{2}} & {(1 + {\hat{ϕ}}_{mle})}^{2}} \end{matrix}) \\ = (\begin{matrix} 2.12 & 2.24 \\ 2.24 & 2.37 \end{matrix}) \end{matrix}$

The proportionality is $\hat{π} = \frac{{\hat{ϕ}}_{mle}}{{\hat{θ}}_{mle}} = 1.053$ , an effect on the spinal curvature due to the adjustments in the angles of sitting on saddle chair. The MS estimates are: $\begin{matrix} \hat{M} S ({\hat{θ}}_{mle}) = \frac{2 {\hat{θ}}_{mle} {\hat{σ}}_{o, x}^{2}}{{(1 + {\hat{θ}}_{mle})}^{2} {\hat{μ}}_{o, x}^{2}} + \\ {{\hat{θ}}_{mle} (1 + {[1 + {\hat{θ}}_{mle}]}^{- 1})}^{2} = 0.221 \end{matrix}$ and $\begin{matrix} \hat{M} S ({\hat{ϕ}}_{mle}) = \frac{2 {\hat{ϕ}}_{mle} {\hat{σ}}_{o, y}^{2}}{{(1 + {\hat{ϕ}}_{mle})}^{2} {\hat{μ}}_{o, y}^{2}} + \\ {({\hat{ϕ}}_{mle} (1 + {[1 + {\hat{ϕ}}_{mle}]}^{- 1}))}^{2} \\ = 0.871 \end{matrix}$

The efficiency

$ɛ_{angle, curvature} = \frac{\hat{MS} ({\hat{ϕ}}_{mle})}{\hat{MS} ({\hat{θ}}_{mle})} = 3.948$ from (3) and (4), we obtain the asymptotic confidence intervals: $- 0.865 < θ < 0.977$

and $- 1.770 < ϕ < 1.888$

However, the estimate of the system’s correlation, using (2), is ${\hat{ρ}}_{mle} = \frac{2.514}{147.058} = 0.131$ , which suggests that the changes in the angle of sitting on saddle chair increases the length of spinal curvature which reduces the back pain. Furthermore, by performing a factor analysis of the output variables, they occur in two clusters: KNM, UB, GLL, and OL are in cluster 1, while GKP. DKP, KLB, and KSM are in cluster 2 (see Fig. 8). It is worth pointing out that the spinal curvature variables UK (place of the maximum deviation of the spinous process on the line) and UL (deviation from the horizontal line of the line joining the inferior angles of the shoulder blades) are insignificant in the context of dimension reduction analysis.

Fig. 7

Sitting angle in saddle versus office chair.

Fig. 8

Clusters of output variables.

4 Limitations and conclusions

The concepts and methods that are developed in this article are limited to the requirement that the data on sitting angles and the length of spinal curvature are Gaussian distributed. In some data collection situations, this requirement might not be valid for a variety of practical reasons. In such a situation, perhaps the data size should be large enough (more than one hundred). With a large sample, even non-Gaussian data could be appropriately transformed to satisfy the Gaussian requirement, according to the well-known central limit theorem. Nevertheless, the findings in this article offers a comfort feeling that sitting on a saddle chair stretches significantly the spinal curvature to reduce back pain if not its total elimination. The physical therapists will be delighted to hear this conclusion based on a statistical analysis of the data and new methodology of this article. No wonder people prefer saddle chair over office chair when it comes to avoid back pain and this article proves the convenience statistically.

Conflict of interest

None to report.

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