help pagetrend

Title

pagetrend -- Page's L trend test for ordered alternatives

Syntax

pagetrend
varlist
[if] [in]

Description

pagetrend is a module to perform Page's (1963) non-parametric test for ordered alternatives.
This test is commonly known as Page's trend test or Page's L test.

The data are required to be in wide form (see help reshape). For example, consider
an experiment in which each of N individuals has been subjected to K different treatments, and the researcher is interested
in the analysis of individual-level treatment outcomes. Then, the data may be organized as:

group x1 x2 ... xK-1 xK
1 3 7 ...	8	6
2 4 3 ...	4	5
... ... ... ... ... ...
N-1 5 5 ...	7	6
N 9 7 ...	8	8

where variable group identifies individuals, and variable xk records individual-specific outcomes under the kth treatment.

Page's trend test is useful when the researcher has a specific hypothesis about the rank ordering of treatments in terms of outcomes
(that is, about which treatments tend to induce greater outcomes). Empirical applications of this test can be found in Abdellaoui, Klibanof and Placido (2013) and Reuben and Riedl (2013) among others.

The researcher's hypothesized rank ordering becomes the alternative hypothesis (e.g. K=4 and Ha: m1 < m2 < m4 < m3 where mk can be the population mean or median of variable xk).
Page's L statistic is based on within-group ranks of treatment outcomes. Given a sample L statistic that exceeds a critical value,
the null (e.g. Ho: m1 = m2 = m4 = m3) would be rejected in favour of the alternative.

The exact distribution of Page's L statistic (L) is a non-standard one. Page (1963, pp.220-221) tabulates a selection of exact critical values. Following Sheskin (2003, p.860),
pagetrend computes an approximate p-value based on an asymptotically standard normal transformation of L: specifically,
based on z = (L - E(L))/sd(L) where E(L) and sd(L) are the expected value and standard deviation of L, respectively.

Note: Cox's (2009) rowranks (findit pr0046)
must be installed before pagetrend can be used. pagetrend calls rowranks to compute within-group ranks,
and specifies method(mean) option of rowranks to address tied ranks.

Example

Continuing with the K=4 example, suppose that the alternative hypothesis is Ha: m1 < m2 < m4 < m3. Then:

. pagetrend x1 x2 x4 x3

Suppose that the alternative hypothesis is Ha: m2 < m4 < m3 < m1 instead. Then:

. pagetrend x2 x4 x3 x1

Note: The variable names do not need to include any common prefix or numeric suffix.

Stored results

pagetrend stores the following in r():

Scalars

r(N_group) number of groups

r(N_treat) number of treatments (i.e. of variables in varlist)

r(L) L statistic

r(z) approximate z statistic (transformation of r(L))

r(p) approximate p-value

Macros

r(treatk) name of the kth variable in varlist

References

Abdellaoui, M., Klibanof P., and Placido, L. 2013. Experiments on compound risk in relation to simple risk and to ambiguity.
Forthcoming in Management Science. [Link]

Cox, N. J. 2009. Speaking Stata: Rowwise. Stata Journal 9(1):
pr0046.

Page, E. B. 1963. Ordered Hypotheses for Multiple Treatments:
A Significance Test for Linear Ranks. Journal of the American Statistical Association 58(301): 216-230.

Reuben, E. and Riedl, A. 2013. Enforcement of contribution norms in public good games with
heterogeneous populations. Games and Economic Behavior 77: 122-137.

Sheskin, D. J. 2003. Handbook of Parametric and Nonparametric Statistical Procedures: Third Edition. CRC Press.

Author

Dr. Hong Il Yoo

Durham University Business School

Durham University

Durham, UK

h.i.yoo@durham.ac.uk

Also see

Help: nptrend, ktau, ranksum, rowranks (if installed), jonter (if installed)