one person takes the measurements (eg length of bone) twice by using same measuring instrument. how can we say that whether the measurements are reproducible or not? is there any statistical test to confirm it ?
If the person goes on to take the measurements of numerous other bones, then you can correlate the first and second measurements as an index of reproducability. There is a statistical test to distinguish that the population correlation is not zero but usually you're just interested in observing a large positive correlation (e.g., > 0.70).
If the person takes only one measurement, but if they measure the bone several times, then you can compute something like a standard deviation (larger means more error). If the number of measurements is not large, you may prefer something like the range used to plot the "whiskers" of a boxplot (which is based on the interquartile range).
If you have several people, you may be able to apply the intraclass correlation coefficient.
But if you have a single person making two measurements of a single bone, you cannot do much.
I will recommend that you use Bland-Altman plots, they are simple to use and interpret. Bland-Altman plots will reveal if reproducibility is the same for all bone lengths. Bland-Altman plots is usually applied when you want to compare two different methods of measurements, but they can be applied in your case.
In general for many cases correlation and regression are important. However, in a comparison of two measurements of the same objects they are pointless.
The correlation coefficient is irrelevant for two reasons.
1) A comparison of methods aims to investigate a systematic difference between two devices or methods. A systematic difference will not affect the correlation coefficient. The variation of each measurements will affect the correlation coefficient, but this is irrelevant when you want to assess the accuracy
2) In case the samples are close to each other (in relation to the analysis uncertainty) they will automatically have a low correlation coefficient. Conversely, you get a high correlation coefficient, if samples which cover a wide range in relation to the analysis uncertainty
Linear regression is problematic for several reasons.
1) Under common linear regression you assume the variables are measured without error (reliably) This is newer the case when you compare methods.
2) Alternatively you can use methods of regression without the assumptions of linear regression. This is complicated statistics (mixed model) and I will presume it's unnecessary in your case.
http://www.biostat.georgiahealth.edu/journalclub/altmanbland1983.pdf
Rune's suggestion that correlations are "pointless" is incorrect. Correlation is a standard method in many areas.
However, Rune raises a point that using correlations as an index of reproducability implies an assumption of random errors. If instead of bone measurements, you were measuring temperature and the first reading was in F and the second in C then the correlation would be close to 1.0 even though the measurements are systematically off. In this case, regression would help you identify both the correlation and the scaling issues. If you just want a way to detect disagreement in general, Cohen's kappa was designed to address agreement, but it is intended for categorical data.
My other suggestions, like using standard deviation to assess the reproducability of a single individual's measurements, are also based on assuming random errors.
I don't understand his second point but there are plenty of other problems that could arise using correlations, such as non-linear relationships. I think these are unlikely in this context but if they did occur then they would preclude regression analysis.
Dear Dr. Laxman.
Intra class correlation coefficient (ICC) can be handy in such situation.
Dr james
can the ICC be used for scale data too ? like i have two value of length of bone (50 cm during first measurement and 50.5 cm in second measurement)?
Yes, you can do it...ICC is applicable for numerical data, be it length of bone or width of teeth. You can check the use of ICC in the following article:
http://bmcoralhealth.biomedcentral.com/articles/10.1186/s12903-016-0265-1
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