Are there formal studies analyzing available test time versus performance? Are there any studies showing that there is a direct connection (or a lack of a connection) between extending the time given for students to complete a written exam and their performance on the exam?

I think there's a lot of anecdotal evidence that would suggest that the more relaxed time constraints are helpful, but I would like to be able to point to "field studies" if possible, as that would make for a more convincing position with my colleagues.
Several studies have examined the impact of extended test-taking time on performance at all academic grade levels. Many of these studies focus on individuals with learning disabilities, but generally include students without learning disabilities as a control group. I have summarized some studies here, and provided a list of additional studies which may be of interest. The studies indicate that other individual factors (academic ability and skills, perception of time, and method of test administration) may impact test performance in concert with or in addition with extended time.

One study of learning disabled (LD) and non-learning disabled (NLD) college students being given a reading test that assessed vocabulary and comprehension assessed outcomes for those who received an extended test period of time and a half compared to the normal period (Ofiesh, 2000). Individuals with LD saw significant improvements in the extended time period compared to NLD students. Some NLD students did show improved scores in the extended period, but the difference was not significant. Similarly, in a study assessing the performance of elementary, middle, and high school students on the Stanford 10 achievement tests who received either the recommended amount of time or extra time, the authors concluded that NLD students exhibited no benefit or detriment with extended testing time (Brooks, Case, &amp; Young, 2003). This did not vary whether they received a few minutes up to double the recommended time.

Another study of 6th grade students taking a standardized test found that NLD students did not see a significant increase in scores in an extended-time condition, but LD students did (Huesman &amp; Frisbie, 2000). However, NLD scores also varied based on instructions. When NLD students were given extra time, and told to take their time, their scores improved; when given extra time but told to work “quickly” but to still do their best work, they did not significantly improve, possibly because they were placed into a mindset of being timed. Another study examined the impact of extended time (1.5) on both multiple choice and problem solving tests in three university courses, but found no improvements in performance (Armitage, 1999). However, in a study of prospective graduate students who took the GRE writing test, participants who were given 60 minutes performed better than those given only 40 minutes (Powers &amp; Fowles, 1996). 

A study of SAT performance for nearly 2,000 LD/ADHD or NLD high school students was examined for groups allowed standard time or 1.5 or 2 times the allotted time for verbal and math sections (Mandibach, Bridgeman, Cahalan-Laitusis, &amp; Trapani, 2005).  They found that NLD medium- and high-ability students performed best in the 1.5 time condition, but low-ability examinees saw no benefit. Extra time affected math more than verbal performance. Strong conclusions about LD student performance were non-significant in part due to small sample sizes. The authors note that low-ability students may not benefit from extra time because they lack needed problem-solving skills, while medium-ability students may benefit because they are able to check their work. Less benefit was seen for high ability students who may not have needed the extra time. The authors also noted that for longer tests, breaking the material into sections and setting time limits per section seemed to benefit all students, rather than asking them to pace themselves across all material. 

One study found no direct effect of extended time on written essay quality for college students, but did find that student who were allowed to use a word processor wrote more than students who provided handwritten answers, and that typed (but not hand written) essays demonstrated a link between length and quality (Lovett, Lewandowski, Berger, &amp; Gathje, 2010). The authors propose it was not time that led to these improvements, but the flexibility to edit their answers when typed.

A review of the evaluation of written exams cites several other studies regarding the impact of time on writing quality (Cho, 2003) and may be of interest. A report by Tindal and Fuchs (2000, pgs 26 – 35) provides a concise summary of the literature of test extensions or untimed tests from kindergarten through post-secondary education, including specific summaries of each study found in their review. This might be a good place to start in terms of additional literature.  Although most of the examples I described here found few effects for non-learning disabled students receiving extended time on tests, I’ve also listed additional literature which may provide alternative results.

Referenced studies:


Armitage, C. (1999). Extended time limits on university examinations (Master’s thesis). Retrieved from: http://dspace.ucalgary.ca/bitstream/1880/24993/1/47990Armitage.pdf
Brooks, T. E., Case, B. J., and Young, M. J. Pearson assessment report. (2003). Timed versus untimed testing conditions and student performance. Pearson Education, Inc. Retrieved from: http://images.pearsonassessments.com/images/tmrs/tmrs_rg/TimedUntimed.pdf
Cho, Y. (2003). Assessing writing: Are we bound by only one method?. Assessing Writing, 8(3), 165-191. doi:10.1016/S1075-2935(03)00018-7
Huesman, R. L., Jr., &amp; Frisbie, D. A. (2000, April). The validity of the ITBS Reading Comprehension test scores for learning disabled and non learning disabled students under extended-time conditions (ED 442 210). Paper presented at the Annual Meeting of the American Education Research Association, New Orleans LA. Retrieved from: http://files.eric.ed.gov/fulltext/ED442210.pdf
Lovett, B. J., Lewandowski, L. J., Berger, C., &amp; Gathje, R. A. (2010). Effects of response mode and time allotment on college students’ writing. Journal of College Reading and Learning, 40(2), 64-79. Retrieved from: http://files.eric.ed.gov/fulltext/EJ887306.pdf
Mandibach, E. B., Bridgeman, B., Cahalan-Laitusis, C., &amp; Trapani, C. (2005). The impact of extended time on SAT test performance. College Board Research Report No. 2005-8 – 35 pages. The College Board; New York. ETS RR-05-20. Retrieved from: https://research.collegeboard.org/sites/default/files/publications/2012/7/researchreport-2005-8-impact-extended-time-sat-test-performance.pdf
Ofiesh, N. S. (2000). Using processing speed tests to predict the benefit of extended test time for university students with learning disabilities. Journal of Postsecondary Education and Disability, 14(1), 39-56. Retrieved from: http://files.eric.ed.gov/fulltext/EJ649045.pdf
Powers, D. E., &amp; Fowles, M. E. (1996). Effects of applying different time limits to a proposed GRE writing test. Journal of Educational Measurement, 33(4), 433-452. doi:10.1111/j.1745-3984.1996.tb00500.x
Tindal, G., &amp;. Fuchs, L. (2000). A summary of research on test changes: An empirical basis for defining accommodations (ED 442 245). Lexington, KY: Mid-South Regional Resource Center. Retrieved from http://files.eric.ed.gov/fulltext/ED442245.pdf


Additional studies:


Alster, E. H. (1997). The effects of extended time on algebra test scores for college students with and without learning disabilities. Journal of Learning Disabilities, 30(2), 222-227. doi:10.1177/002221949703000210
Calahan-Laitusis, C. (2004). Accommodations on high-stakes writing tests for students with disabilities. Princeton, NJ: ETS. Retrieved from: http://144.81.87.152/Media/Research/pdf/RR-04-13.pdf
Cohen, A. S., Gregg, N., &amp; Deng, M. (2005). The Role of Extended Time and Item Content on a High‐Stakes Mathematics Test. Learning Disabilities Research &amp; Practice, 20(4), 225-233. doi:10.1111/j.1540-5826.2005.00138.x
Crawford, L., Helwig, R., &amp; Tindal, G. (2004). Writing Performance Assessments How Important Is Extended Time?. Journal of learning disabilities, 37(2), 132-142. doi:10.1177/00222194040370020401
Knoch, U., &amp; Elder, C. (2010). Validity and fairness implications of varying time conditions on a diagnostic test of academic English writing proficiency. System,38(1), 63-74. doi:10.1016/j.system.2009.12.006
Lewandowski, L. J., Lovett, B. J., Parolin, R., Gordon, M., &amp; Codding, R. S. (2007). Extended time accommodations and the mathematics performance of students with and without ADHD. Journal of Psychoeducational Assessment,25(1), 17-28. doi:10.1177/0734282906291961
Lewandowski, L. J., Lovett, B. J., &amp; Rogers, C. L. (2008). Extended Time as a Testing Accommodation for Students With Reading Disabilities Does a Rising Tide Lift All Ships?. Journal of Psychoeducational Assessment, 26(4), 315-324. doi:10.1177/0734282908315757
Munger, G. F., &amp; Loyd, B. H. (1991). Effect of speededness on test performance of handicapped and nonhandicapped examinees. The Journal of Educational Research, 85(1), 53-57. doi:10.1080/00220671.1991.10702812
Runyan, M. K. (1991). The effect of extra time on reading comprehension scores for university students with and without learning disabilities. Journal of Learning Disabilities, 24(2), 104-108. doi:10.1177/002221949102400207