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Ton J. Cleophas and Aeilko H. ZwindermanSPSS for Starters and 2nd Levelers10.1007/978-3-319-20600-4_25

25. Curvilinear Estimation (20 Patients)

Ton J. Cleophas^{1, 2} and Aeilko H. Zwinderman^2,
3

(1)

Department Medicine, Albert Schweitzer Hospital, Dordrecht, The Netherlands

(2)

European College Pharmaceutical Medicine, Lyon, France

(3)

Department Biostatistics, Academic Medical Center, Amsterdam, The Netherlands

1 General Purpose

The general principle of regression analysis is that the best fit line/exponential-curve/curvilinear-curve etc. is calculated, i.e., the one with the shortest distances to the data, and that it is, subsequently, tested how far the data are from the curve. A significant correlation between the y (outcome data) and the x (exposure data) means that the data are closer to the model than will happen purely by chance. The level of significance is usually tested, simply, with t-tests or analysis of variance. The simplest regression model is a linear model.

2 Schematic Overview of Type of Data File

3 Primary Scientific Question

Is curvilinear regression able to find a best fit regression model for data with both a continuous outcome and predictor variable.

4 Data Example

In a 20 patient study the quantity of care estimated as the numbers of daily interventions like endoscopies and small operations per doctor is tested against the quality of care scores. The primary question was: if the relationship between quantity of care and quality of care is not linear, does curvilinear regression help find the best fit curve?

Quantityscore	Qualityscore
19,00	2,00
20,00	3,00
23,00	4,00
24,00	5,00
26,00	6,00
27,00	7,00
28,00	8,00
29,00	9,00
29,00	10,00
29,00	11,00

quantityscore quantity of care (numbers of daily intervention per doctor)

qualityscore quality of care scores.

The first ten patients of the data file is given above. The entire data file is in extras.springer.com, and is entitled chapter25curvilinearestimation. Start by opening that data file in SPSS. First, we will make a graph of the data.

5 Data Graph

Command:

Analyze….Graphs….Chart builder….click: Scatter/Dot….click quality of care and drag to the Y-Axis….click interventions per doctor and drag to the X-Axis….click OK.

The above graph shows the scattergram of the data. A nonlinear relationship is suggested. The curvilinear regression option in SPSS helps us identify the best fit model.

6 Curvilinear Estimation

For analysis, the statistical model Curve Estimation in the module Regression is required.

Command:

Analyze….Regression….Curve Estimation….mark: Linear, Logarithmic, Inverse, Quadratic, Cubic, Power, Exponential….mark: Display ANOVA Table….click OK.

The above graph is produced by the software program. It looks as though the quadratic and cubic models produce the best fit models. All of the curves are tested for goodness of fit using analysis of variance (ANOVA). The underneath tables show the calculated B-values (regression coefficients). The larger the absolute B-values, the better fit is provided by the model. The tables also test whether the absolute B-values are significantly larger than 0,0. 0,0 indicates no relationship at all. Significantly larger than 0,0 means, that the data are closer to the curve than could happen by chance. The best fit linear, logarithmic, and inverse models are not statistically significant. The best fit quadratic and cubic models are very significant. The power models and exponential models are, again, not statistically significant.

Coefficients

	Unstandardized coefficients		Standardized coefficients	t	Sig.
	B	Std. error	Beta	t	Sig.
Interventions/doctor	−,069	,116	−,135	−,594	,559
(Constant)	25,588	1,556		16,440	,000

(1) Linear

Coefficients

	Unstandardized coefficients		Standardized coefficients	t	Sig.
	B	Std. error	Beta	t	Sig.
In(interventions/doctor)	,726	1,061	,155	,684	,502
(Constant)	23,086	2,548		9,061	,000

(2) Logarithmic

Coefficients

	Unstandardized coefficients		Standardized coefficients	t	Sig.
	B	Std. error	Beta	t	Sig.
1/interventions/doctor	−11,448	5,850	−,410	−1,957	,065
(Constant)	26,229	,989		26,512	,000

(3) Inverse

Coefficients

	Unstandardized coefficients		Standardized coefficients	t	Sig.
	B	Std. error	Beta	t	Sig.
Interventions/doctor	−2,017	,200	3,960	10,081	,000
Interventions/doctor**2	−,087	,008	−4,197	−10,686	,000
(Constant)	16,259	1,054		15,430	,000

(4) Quadratic

Coefficients

	Unstandardized coefficients		Standardized coefficients	t	Sig.
	B	Std. error	Beta	t	Sig.
Interventions/doctor	4,195	,258	8,234	16,234	,000
Interventions/doctor**2	−,301	,024	−14,534	−12,437	,000
Interventions/doctor**3	,006	,001	6,247	8,940	,000
(Constant)	10,679	,772		13,836	,000

(5) Cubic

Coefficients

	Unstandardized coefficients		Standardized coefficients	t	Sig.
	B	Std. error	Beta	t	Sig.
In(interventions/doctor)	,035	,044	,180	,797	,435
(Constant)	22,667	2,379		9,528	,000

The dependent variable is ln (qual care score)

(6) Power

Coefficients

	Unstandardized coefficients		Standardized coefficients	t	Sig.
	B	Std. error	Beta	t	Sig.
Interventions/doctor	−,002	,005	−,114	−,499	,624
(Constant)	25,281	1,632		15,489	,000

The dependent variable is ln (qual care score)

(7) Exponential

The largest test statistics are given by (4) Quadratic and (5) Cubic. Now, we can construct regression equations for these two best fit curves using the data from the ANOVA tables.

(4) Quadratic

$\begin{array}{ll}\mathrm{y}=\mathrm{a}+\mathrm{b}\mathrm{x}+{\mathrm{cx}}^2\hfill & =16.259+2.017\mathrm{x}\hbox{--} 0.087{\mathrm{x}}^2\hfill \\ {}\hfill & =16.3+2.0\mathrm{x}\hbox{--} 0.09{\mathrm{x}}^2\hfill \end{array}$

(5) Cubic

$\begin{array}{ll}\mathrm{y}=\mathrm{a}+\mathrm{b}\mathrm{x}+{\mathrm{cx}}^2+{\mathrm{dx}}^3\hfill & =10.679+4.195\mathrm{x}-0.301{\mathrm{x}}^2+0.006{\mathrm{x}}^3\hfill \\ {}\hfill & =10.7+4.2\mathrm{x}-0.3{\mathrm{x}}^2+0.006{\mathrm{x}}^3\hfill \end{array}$

The above equations can be used to make a prediction about the best fit y-value from a given x-value, e.g., with x = 10 you might expect an y-value of

$\begin{array}{l}\mathrm{y}=16.3+20-9=27.3\ \mathrm{according}\ \mathrm{t}\mathrm{o}\ \mathrm{t}\mathrm{he}\ \mathrm{quadratic}\ \mathrm{model}\hfill \\ {}\mathrm{y}=10.7+42\hbox{--} 30+6=28.7\ \mathrm{according}\ \mathrm{t}\mathrm{o}\ \mathrm{t}\mathrm{he}\ \mathrm{cubic}\ \mathrm{model}.\hfill \end{array}$

Alternatively, predictions about the best fit y-values from x-values given can also be fairly accurately extrapolated from the curves as drawn.

7 Conclusion

The relationship between quantity of care and quality of care is curvilinear. Curvilinear regression has helped finding the best fit curve. If the standard curvilinear regression models do not yet fit the data, then there are other possibilities, like logit and probit transformations, Box Cox transformations, ACE (alternating conditional expectations)/AVAS (additive and variance stabilization) packages, Loess (locally weighted scatter plot smoothing) and spline modeling (see also Chap. 26). These methods are, however, increasingly complex, and, often, computationally very intensive. But, for a computer this is no problem.

8 Note

More background, theoretical, and mathematical information of curvilinear estimation is given in Statistics applied to clinical studies 5th edition, Chaps. 16 and 24, Springer Heidelberg Germany, 2012, from the same authors.

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