ELDA: Extreme Limiting Dilution Analysis


This page performs limiting dilution analysis using ELDA software. ELDA is suitable for any limiting dilution problem, but special methods are implemented to give reliable results in extreme data situations, for example when all the assays give positive or negative results. It is especially suitable for analysing limiting dilution data arising in stem cell research.

ELDA can compare multiple groups or treatment conditions and can test for goodness of fit.

Users of the R statistical programming environment can access the command-line version of ELDA through the limdil function of the statmod package.

This page replaces the limdil webpage which was formerly at http://bioinf.wehi.edu.au/software/limdil.

How to cite ELDA

Please cite as:

Hu, Y, and Smyth, GK (2009). ELDA: Extreme limiting dilution analysis for comparing depleted and enriched populations in stem cell and other assays. Journal of Immunological Methods 347, 70-78.

The full text of the article can be downloaded from

Data Entry

Data should be input as a data table of three or four columns, separated by any combination of commas, spaces or tabs. The data can be typed directly into the text field below, or can be cut and pasted from any spreadsheet application. Each row of data gives results for a particular cell dose. The columns are:

  1. Dose: number of cells in each culture
  2. Tested: number of cultures tested
  3. Response: number of positive cultures
  4. Group (optional): label for the population group to which cells belong

Enter your data in this box:

 Required confidence interval
 Plot results? Legend position:  bottomleft?  topright?  topleft?  bottomright?
 Test for inequality in frequency between multiple groups?
 Test adequacy of the single-hit model?
 Is the actual number of cells observed?


A single group analysis:

Enter the following input, then select the model test check box, but not the observed check box.

50 24 2
100 24 6
200 24 9

Confidence intervals for 1/(stem cell frequency).
Lower Estimate Upper
538 403 302

Goodness of fit tests. These test whether the log-dose slope equals 1. Rejection of the tests may be due either to batch effects (heterogeneity in the stem cell frequencies or assay success rate) or to a failure of the stem cell hypothesis.
Estimated slope is 1.05
Test Chisq DF P Value
Likelihood ratio test of single-hit model 0.0667 1 0.796
Score test of heterogeneity 0.0509 1 0.822

A multiple groups analysis:

Enter the following input, then select the single hit model test check box and the multiple groups check box but not the observed check box.

30000 6 2 A
20000 6 3 A
4000 6 2 A
500 6 1 A
30000 6 6 B
20000 6 5 B
4000 6 6 B
500 6 1 B
30000 6 2 C
20000 6 3 C
4000 6 4 C
500 6 2 C

Confidence intervals for 1/(stem cell frequency)
Group Lower Estimate Upper
A 67165 31864 15116
B 10252 4509 1983
C 44526 22491 11361

Overall test for differences in stem cell frequencies between any of the groups.
Chisq DF P.value
18.1 2 0.000119

Pairwise tests for differences in stem cell frequencies.
Group 1 Group 2 Chisq DF Pr(>Chisq)
A B 16.7 1 4.38e-05
A C 0.549 1 0.459
B C 12.4 1 0.000432

Goodness of fit tests. These test whether the log-dose slope equals 1. Rejection of the tests may be due either to batch effects (heterogeneity in the stem cell frequencies or assay success rate) or to a failure of the stem cell hypothesis.
Estimated slope is 0.293
Test Chisq DF P Value
Likelihood ratio test of single-hit model 24.4 1 7.73e-07
Score test of heterogeneity 44.7 1 2.25e-11

Interpreting goodness of fit tests from ELDA

The likelihood ratio test is designed to test whether the single-hit model is correct.

The score test is designed to test whether the different cultures (assays) have the same active cell proportion. (For example, it is quite likely in practice that experiments done at different times might give rise to slightly different stem cell frequencies, and this test will respond to that.)

The nominal value of the slope is 1. A slope greater than one suggests a multi-hit model in which two or more cells are synergistically required to produce a positive response. A slope less than 1 suggests some sort of cell interference. Slopes less than 1 can also be due to heterogeneity.

Selected publications using ELDA

  • Asselin-Labat, M.L., Vaillant, F., Sheridan, J.M., Pal, B., Wu, D., Simpson, E.R., Yasuda, H., Smyth, G.K., Martin, T.J., Lindeman, G.J., Visvader, J.E., (2010). Control of mammary stem cell function by steroid hormone signalling. Nature 465, 798.
  • Atkinson, R.L., Zhang, M., Diagaradjane, P., Peddibhotla, S., Contreras, A., Hilsenbeck, S.G., Woodward, W.A., Krishnan, S., Chang, J.C., Rosen, J.M., (2010). Thermal enhancement with optically activated gold nanoshells sensitizes breast cancer stem cells to radiation therapy. Sci Transl Med. 55, 55ra79.
  • Cejka, P., Cannavo, E., Polaczek, P., Masuda-Sasa, T., Pokharel, S., Campbell, J.L., Kowalczykowski, S.C., (2010). DNA end resection by Dna2-Sgs1-RPA and its stimulation by Top3-Rmi1 and Mre11-Rad50-Xrs2. Nature 467, 112.
  • Diaz-Guerra, E., Vernal, R., del Prete, M.J., Silva, A., Garcia-Sanz, J.A., (2007). CCL2 inhibits the apoptosis program induced by growth factor deprivation, rescuing functional T cells. J. Immunol. 179, 7352.
  • Eirew, P., Stingl, J., Eaves, C.J., (2010). Quantitation of human mammary epithelial stem cells with in vivo regenerative properties using a subrenal capsule xenotransplantation assay. Nat Protoc. 5, 1945.
  • Gracheva, E.O., Ingolia, N.T., Kelly, Y.M., Cordero-Morales, J.F., Hollopeter, G., Chesler, A.T., S├íhez, E.E., Perez, J.C., Weissman, J.S., Julius, D., (2010). Molecular basis of infrared detection by snakes. Nature 464, 1006.
  • Holst, J., Watson, S., Lord, M.S., Eamegdool, S.S., Bax, D.V., Nivison-Smith, L.B., Kondyurin, A., Ma, L., Oberhauser, A.F., Weiss, A.S., Rasko, J.E., (2010). Substrate elasticity provides mechanical signals for the expansion of hemopoietic stem and progenitor cells. Nat Biotechnol. 28, 1123.
  • Hosen, N., Yamane, T.,Muijtjens,M., Pham, K., Clarke, M.F., Weissman, I.L., (2007). Bmi-1-green fluorescent protein-knock-in mice reveal the dynamic regulation of bmi-1 expression in normal and leukemic hematopoietic cells. Stem Cells 25, 1635.
  • Lancrin, C., Sroczynska, P., Stephenson, C., Allen, T., Kouskoff, V., Lacaud, G., (2009). The haemangioblast generates haematopoietic cells through a haemogenic endothelium stage. Nature 457, 892.
  • Leong, K.G., Wang, B.E., Johnson, L., Gao, W.Q., (2008). Generation of a prostate from a single adult stem cell. Nature 456, 804.
  • Notta, F., Doulatov, S., Laurenti, E., Poeppl, A., Jurisica, I., Dick, J.E., (2011). Isolation of single human hematopoietic stem cells capable of long-term multilineage engraftment. Science 333, 218.
  • Quintana, E., Shackleton,M., Sabel, M.S., Fullen, D.R., Johnson, T.M.,Morrison, S.J., (2008). Efficient tumour formation by single human melanoma cells. Nature 456, 593.
  • Shackleton, M., Vaillant, F., Simpson,K.J., Stingl, J., Smyth, G.K., Asselin-Labat,M.-L., Wu, L., Lindeman, G.J., Visvader, J.E., (2006). Generation of a functional mammary gland from a single stem cell. Nature 439, 84.
  • Siwko, S.K., Dong, J., Lewis, M.T., Liu, H., Hilsenbeck, S.G., Li, Y., (2008). Evidence that an early pregnancy causes a persistent decrease in the number of functional mammary epithelial stem cells-implications for pregnancy- induced protection against breast cancer. Stem Cells 26, 3205.
  • Vaillant, F., Asselin-Labat, M.L., Shackleton, M., Forrest, N.C., Lindeman, G.J., Visvader, J.E., (2008). The mammary progenitor marker CD61/B3 integrin identifies cancer stem cells in mouse models of mammary tumorigenesis. Cancer Res. 68, 7711.
  • Vermeulen, L., Todaro, M., de sousa Mello, F., Sprick, M.R., Kemper, K., Perez Alea, M., Richel, D.J., Stassi, G., Medema, J.P., (2008). Single-cell cloning of colon cancer stem cells reveals a multi-lineage differentiation capacity. Proc. Natl. Acad. Sci. U. S. A. 105, 13427.

Comments/Questions? Email <Yifang Hu>

Last Modified: 24 October 2014