hw8 and task 2

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ManualregressioncalculationofMotulskyexamplech.22-23updated.xlsx

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Background

Motulsky example for correlation and linare regression (chapters 22 and 23)

lipids and insulin sensitivity

Experiment:

select random, healthy, male subjects.

infuse insulin at a standard rate.

infuse glucose to maintain blood glucose level constant

This determines insulin sensitivity as follows:

insulin triggers cells to take up glucose.

the higher someone's insulin sensitivity, the more glucose is taken up at the same insulin level.

Thus, the speed of glucose uptake (in mg/m2/min) is an indirect measure for insulin sensitivity.

Total glucose uptake (as measured from declining levels in the blood) is also dependent on the size and weight of an individual.

Therefore, the speed of glucose uptake is standardized. This is done not by weight but by body surface (in m2).

Determine the fatty acid composition of muscle cells (cell membranes) from biopsies.

Specifically, analyze the proportion of C20-C22 polyunsaturated fatty acids within all fatty acids.

Raw data

	%C20-22 fatty acids	Insulin sensitivity
	polyunsaturated	[mg/m2/min]
	17.9	250
	18.3	220
	18.3	145
	18.4	115
	18.4	230
	20.2	200
	20.3	330
	21.8	400
	21.9	370
	22.1	260
	23.1	270
	24.2	530
	24.4	375

Correlation between insulin sensitivity and fatty acid content

17.899999999999999 18.3 18.3 18.399999999999999 18.399999999999999 20.2 20.3 21.8 21.9 22.1 23.1 24.2 24.4 250 220 145 115 230 200 330 400 370 260 270 530 375

%C20-22 Fatty Acids

Insulin sensitivity

[mg/m2/min]

lines of best fit

Correlation between insulin sensitivity and fatty acid content

17.899999999999999 18.3 18.3 18.399999999999999 18.399999999999999 20.2 20.3 21.8 21.9 22.1 23.1 24.2 24.4 250 220 145 115 230 200 330 400 370 260 270 530 375

%C20-22 Fatty Acids

Insulin sensitivity

[mg/m2/min]

calculation of R, m, and b

	%C20-22	Insulin sens.
	fatty acids	[mg/m2/min]		distance to mean			Product of distances		(distance of x)2
	X	Y							(for slope calculation)
	17.9	250		-2.815	-34.231		96.373		7.926
	18.3	220		-2.415	-64.231		155.142		5.834
	18.3	145		-2.415	-139.231		336.296		5.834
	18.4	115		-2.315	-169.231		391.834		5.361
	18.4	230		-2.315	-54.231		125.565		5.361
	20.2	200		-0.515	-84.231		43.411		0.266
	20.3	330		-0.415	45.769		-19.012		0.173
	21.8	400		1.085	115.769		125.565		1.176
	21.9	370		1.185	85.769		101.604		1.403
	22.1	260		1.385	-24.231		-33.550		1.917
	23.1	270		2.385	-14.231		-33.935		5.686
	24.2	530		3.485	245.769		856.411		12.143
	24.4	375		3.685	90.769		334.450		13.576
Average	20.715	284.231					2480.154		66.657
Std.Dev.S	2.357	113.887
Std.Dev.P	2.264	109.419
							r	r2
							0.834	0.6958
						Motulsky r2		0.5929
							0.7700025428	0.5929	OK. So they use the sample standard deviation and not the population standard deviation!
Calculation of slope (I call it m the formla calls it b) and intercept (I call it b, the formula calls it a)
				2480.1538461538
				66.6569230769		b=	37.2077457475	slope
						a=	-486.5419945992	intercept
		284.2307692308	20.7153846154
		y avg.	x avg.

calculating SS and MS

									mean of y		284.2307692308	(from cell C26)
									y' value of regression line: y'=bx+a			with	b=	37.2077457475	slope	from calcs. in previous tab
													a=	-486.5419945992	intercept	from calcs. in previous tab
	%C20-22	Insulin sens.
	fatty acids	[mg/m2/min]
	X	Y	Y total	Ytotal2	Y regression	Yregr.2	y Residual	Yresid.2
	17.9	250	-34.2307692308	1171.7455621302	-104.7541149505	10973.4245990685	70.5233457198	4973.5422915087
	18.3	220	-64.2307692308	4125.5917159763	-89.8710166515	8076.7996339825	25.6402474208	657.4222877987
	18.3	145	-139.2307692308	19385.2071005917	-89.8710166515	8076.7996339825	-49.3597525792	2436.3851746821
	18.4	115	-169.2307692308	28639.0532544379	-86.1502420768	7421.8642098913	-83.080527154	6902.3739921813
	18.4	230	-54.2307692308	2940.976331361	-86.1502420768	7421.8642098913	31.919472846	1018.8527467685
	20.2	200	-84.2307692308	7094.8224852071	-19.1762997314	367.7304713878	-65.0544694994	4232.0840018468
	20.3	330	45.7692307692	2094.8224852071	-15.4555251566	238.8732578674	61.2247559259	3748.4707381819
	21.8	400	115.7692307692	13402.5147928994	40.3560934645	1628.6142797193	75.4131373047	5687.141278135
	21.9	370	85.7692307692	7356.3609467456	44.0768680393	1942.7702961533	41.6923627299	1738.2531100047
	22.1	260	-24.2307692308	587.1301775148	51.5184171888	2654.1473096376	-75.7491864196	5737.9392432243
	23.1	270	-14.2307692308	202.5147928994	88.7261629362	7872.3319893882	-102.956932167	10600.1298812421
	24.2	530	245.7692307692	60402.5147928994	129.6546832584	16810.3368908466	116.1145475108	13482.5881436355
	24.4	375	90.7692307692	8239.0532544379	137.0962324079	18795.3769404496	-46.3270016387	2146.1910808322
SUM OF SQUARES				155642.307692308		92280.933722266		63361.3739700418		Results from Minitab Fitted Line Plot analysis
Average	20.715	284.231									df	SS	MS
Std.Dev.S	2.357	113.887								Regression	1	92280.9337222659	92280.9337222659
										Residual	11	63361.3739700418	5760.1249063674
										Total	12	155642.307692308
					The "standard deviation"
					of the Sum of Squares (residuals)
					is called the Root Mean Square (RMS) error and
					is calculated like SD, but with N-2 in the denominator
					Sum of squares of residuals:			63361.3739700418		(from above)
					Divide by N-2 with N=13			5760.1249063674		(see value for MS(residual) in minitab output above)
					take the square root of that			75.8954867325		Standard error of residuals
					this is the error displayed in minitab and Excel outputs!
					compare to standard error shown in Motulsky textbook, table 23.1 p. 153
					(standard error is called Sy,x or standard deviation of the residuals

p value for Regression

CALCULATION OF P VALUE FOR RESIDUALS USING VARIANCE ANALYSIS AND F DISTRIBUTION

confidence interval for R

CONFIDENCE INTERVAL FOR PEARSON CORRELATION COEFFICIENT R
(not given by minitab or Excel)
	INPUT
r=	0.7700025428
N=	13
α =	0.05
Step 1: Transform r value to z value, using Fisher r to z transformation
Formula:			EXCEL function		FISHER(x)
			(replace x with your r value)
					OUTPUT
					Fisher (B4)
Transform r from above into z:		Calculation:		z=	1.0203340046
Step 2: Calculate the confidence interval for the transformed z value
The z distribution for r values is a normal distribution,
but with a standard deviation of
instead of 1.
For confidence level of 95%:					OUTPUT
Lower confidence limit for z:		Formula	zlow = z - 1.96 * SD = z - 1.96 / sqrt (N-3)
		Calculation	zlow =		0.4005275832
Upper confidence limit for z:		Formula	zhigh = z + 1.96 * SD = z + 1.96 / sqrt (N-3)
		Calculation	zhigh =		1.640140426
Step 3: do a backward transformation from z to r to calculate the confidence interval for r
Formula for inverse Fisher r to z transformation (i.e. Fisher z to r transformation)
			EXCEL function:		FISHERINV(y)
					OUTPUT
For confidence level of 95%:					FISHERINV(G25)
Lower confidence limit for r:			rlow =		0.3804002924
					FISHERINV(G28)
Upper confidence limit for r:			rhigh =		0.9274921955