# [R] help in interpreting paired t-test

Marc Schwartz marc_schwartz at me.com
Wed Sep 21 18:39:34 CEST 2011

```Jeremy,

Correlation alone is irrelevant when comparing two separate sets of measurements on the same specimen. Correlation does not mean good agreement, but good agreement tends to infer high correlation.

T1 <- rnorm(50, mean = 100)

> mean(T1)
[1] 99.80257

T2 <- T1 * 1.5

> mean(T2)
[1] 149.7039

The two measures are off by a systematic 50%, but:

> cor(T1, T2)
[1] 1

The key here, as I noted in my reply yesterday and as Greg noted in his this morning regarding Bland-Altman, is whether or not the two measures agree within an acceptable margin of error and whether or not there is systematic bias in the measures, either overall or perhaps one measure tends to be low at one end of the range, while high at the other.

HTH,

Marc Schwartz

On Sep 21, 2011, at 11:20 AM, Jeremy Miles wrote:

>> cor(A, B)
> [1] 0.9986861
>
> The data are very, very highly correlated. The higher the correlation,
> the greater the power of the t-test to detect the same difference
> between the means.
>
> Jeremy
>
> On 20 September 2011 10:46, Pedro Mardones <mardones.p at gmail.com> wrote:
>> Dear all;
>>
>> A very basic question. I have the following data:
>>
>> ************************************************************************************
>>
>> A <- 1/1000*c(347,328,129,122,18,57,105,188,57,257,53,108,336,163,
>> 62,112,334,249,45,244,211,175,174,26,375,346,153,32,
>> 89,32,358,202,123,131,88,36,30,67,96,135,219,122,
>> 89,117,86,169,179,54,48,40,54,568,664,277,91,290,
>> 116,80,107,401,225,517,90,133,36,50,174,103,192,150,
>> 225,29,80,199,55,258,97,109,137,90,236,109,204,160,
>> 95,54,50,78,98,141,508,144,434,100,37,22,304,175,
>> 72,71,111,60,212,73,50,92,70,148,28,63,46,85,
>> 111,67,234,65,92,59,118,202,21,17,95,86,296,45,
>> 139,32,21,70,185,172,151,129,42,14,13,75,303,119,
>> 128,106,224,241,112,395,78,89,247,122,212,61,165,30,
>> 65,261,415,159,316,182,141,184,124,223,39,141,103,149,
>> 104,71,259,86,85,214,96,246,306,11,129)
>>
>> B <- 1/1000*c(351,313,130,119,17,50,105,181,58,255,51,98,335,162,
>> 60,108,325,240,44,242,208,168,170,27,356,341,150,31,
>> 85,29,363,185,124,131,85,35,27,63,92,147,217,117,
>> 87,119,81,161,178,53,45,38,50,581,661,254,87,281,
>> 110,76,100,401,220,507,94,123,36,47,154,99,184,146,
>> 232,26,77,193,53,264,94,110,128,87,231,110,195,156,
>> 95,51,50,75,93,134,519,139,435,96,37,21,293,169,
>> 70,80,104,64,210,70,48,88,67,140,26,52,45,90,
>> 106,63,219,62,91,56,113,187,18,14,95,86,284,39,
>> 132,31,22,69,181,167,150,117,42,14,11,73,303,109,
>> 129,106,227,249,111,409,71,88,256,120,200,60,159,27,
>> 63,268,389,150,311,175,136,171,116,220,30,145,95,148,
>> 102,70,251,88,83,199,94,245,305,9,129)
>>
>> ************************************************************************************
>>
>> plot(A,B)
>> abline(0,1)
>>
>> At a glance, the data look very similar. Data A and B are two
>> measurements of the same variable but using different devices (on a
>> same set of subjects). Thus, I thought that a paired t-test could be
>> appropriate to check if the diff between measurement devices = 0.
>>
>> t.test(A-B)
>>
>> ************************************************************************************
>>
>> One Sample t-test
>>
>> data:  A - B
>> t = 7.6276, df = 178, p-value = 1.387e-12
>> alternative hypothesis: true mean is not equal to 0
>> 95 percent confidence interval:
>>  0.002451622 0.004162903
>> sample estimates:
>>  mean of x
>> 0.003307263
>>
>> ************************************************************************************
>> The mean diff is 0.0033 but the p-value indicates a strong evidence to
>> reject H0.
>>
>> I was expecting to find no differences so I'm wondering whether the
>> t-test is the appropriate test to use. I'll appreciate any comments or
>> suggestions.
>>
>> BR,
>> PM
>>
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